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Theorem infxpenlem 9122
Description: Lemma for infxpen 9123. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
leweon.1 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
r0weon.1 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
infxpen.1 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
infxpen.2 (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))
infxpen.3 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
infxpen.4 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
Assertion
Ref Expression
infxpenlem ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Distinct variable groups:   𝐴,𝑎   𝑤,𝐽   𝑧,𝑤,𝐿   𝑧,𝑚,𝑀   𝜑,𝑤,𝑧   𝑧,𝑄   𝑚,𝑎,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚,𝑎)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑚)   𝑄(𝑥,𝑦,𝑤,𝑚,𝑎)   𝑅(𝑥,𝑦,𝑧,𝑤,𝑚,𝑎)   𝐽(𝑥,𝑦,𝑧,𝑚,𝑎)   𝐿(𝑥,𝑦,𝑚,𝑎)   𝑀(𝑥,𝑦,𝑤,𝑎)

Proof of Theorem infxpenlem
StepHypRef Expression
1 sseq2 3831 . . . 4 (𝑎 = 𝑚 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝑚))
2 xpeq12 5342 . . . . . 6 ((𝑎 = 𝑚𝑎 = 𝑚) → (𝑎 × 𝑎) = (𝑚 × 𝑚))
32anidms 558 . . . . 5 (𝑎 = 𝑚 → (𝑎 × 𝑎) = (𝑚 × 𝑚))
4 id 22 . . . . 5 (𝑎 = 𝑚𝑎 = 𝑚)
53, 4breq12d 4864 . . . 4 (𝑎 = 𝑚 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝑚 × 𝑚) ≈ 𝑚))
61, 5imbi12d 335 . . 3 (𝑎 = 𝑚 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)))
7 sseq2 3831 . . . 4 (𝑎 = 𝐴 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝐴))
8 xpeq12 5342 . . . . . 6 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑎 × 𝑎) = (𝐴 × 𝐴))
98anidms 558 . . . . 5 (𝑎 = 𝐴 → (𝑎 × 𝑎) = (𝐴 × 𝐴))
10 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
119, 10breq12d 4864 . . . 4 (𝑎 = 𝐴 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝐴 × 𝐴) ≈ 𝐴))
127, 11imbi12d 335 . . 3 (𝑎 = 𝐴 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)))
13 infxpen.2 . . . . . . . 8 (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))
14 vex 3401 . . . . . . . . . . . . 13 𝑎 ∈ V
1514, 14xpex 7195 . . . . . . . . . . . 12 (𝑎 × 𝑎) ∈ V
16 simpll 774 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
1713, 16sylbi 208 . . . . . . . . . . . . . . . . 17 (𝜑𝑎 ∈ On)
18 onss 7223 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ On → 𝑎 ⊆ On)
1917, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑎 ⊆ On)
20 xpss12 5333 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ On ∧ 𝑎 ⊆ On) → (𝑎 × 𝑎) ⊆ (On × On))
2119, 19, 20syl2anc 575 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 × 𝑎) ⊆ (On × On))
22 leweon.1 . . . . . . . . . . . . . . . . 17 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
23 r0weon.1 . . . . . . . . . . . . . . . . 17 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
2422, 23r0weon 9121 . . . . . . . . . . . . . . . 16 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
2524simpli 472 . . . . . . . . . . . . . . 15 𝑅 We (On × On)
26 wess 5305 . . . . . . . . . . . . . . 15 ((𝑎 × 𝑎) ⊆ (On × On) → (𝑅 We (On × On) → 𝑅 We (𝑎 × 𝑎)))
2721, 25, 26mpisyl 21 . . . . . . . . . . . . . 14 (𝜑𝑅 We (𝑎 × 𝑎))
28 weinxp 5395 . . . . . . . . . . . . . 14 (𝑅 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
2927, 28sylib 209 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
30 infxpen.1 . . . . . . . . . . . . . 14 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
31 weeq1 5306 . . . . . . . . . . . . . 14 (𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) → (𝑄 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎)))
3230, 31ax-mp 5 . . . . . . . . . . . . 13 (𝑄 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
3329, 32sylibr 225 . . . . . . . . . . . 12 (𝜑𝑄 We (𝑎 × 𝑎))
34 infxpen.4 . . . . . . . . . . . . 13 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
3534oiiso 8684 . . . . . . . . . . . 12 (((𝑎 × 𝑎) ∈ V ∧ 𝑄 We (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
3615, 33, 35sylancr 577 . . . . . . . . . . 11 (𝜑𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
37 isof1o 6800 . . . . . . . . . . 11 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎))
38 f1ocnv 6368 . . . . . . . . . . 11 (𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎) → 𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽)
39 f1of1 6355 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
4036, 37, 38, 394syl 19 . . . . . . . . . 10 (𝜑𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
41 f1f1orn 6367 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽)
4215f1oen 8216 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽 → (𝑎 × 𝑎) ≈ ran 𝐽)
4340, 41, 423syl 18 . . . . . . . . 9 (𝜑 → (𝑎 × 𝑎) ≈ ran 𝐽)
44 f1ofn 6357 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽 Fn (𝑎 × 𝑎))
4536, 37, 38, 444syl 19 . . . . . . . . . 10 (𝜑𝐽 Fn (𝑎 × 𝑎))
4636adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
4737, 38, 393syl 18 . . . . . . . . . . . . . . . . . 18 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
48 cnvimass 5702 . . . . . . . . . . . . . . . . . . 19 (𝑄 “ {𝑤}) ⊆ dom 𝑄
49 inss2 4037 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
5030, 49eqsstri 3839 . . . . . . . . . . . . . . . . . . . . 21 𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
51 dmss 5531 . . . . . . . . . . . . . . . . . . . . 21 (𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎)) → dom 𝑄 ⊆ dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 dom 𝑄 ⊆ dom ((𝑎 × 𝑎) × (𝑎 × 𝑎))
53 dmxpid 5553 . . . . . . . . . . . . . . . . . . . 20 dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)) = (𝑎 × 𝑎)
5452, 53sseqtri 3841 . . . . . . . . . . . . . . . . . . 19 dom 𝑄 ⊆ (𝑎 × 𝑎)
5548, 54sstri 3814 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)
56 f1ores 6370 . . . . . . . . . . . . . . . . . 18 ((𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽 ∧ (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5747, 55, 56sylancl 576 . . . . . . . . . . . . . . . . 17 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5815, 15xpex 7195 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 × 𝑎) × (𝑎 × 𝑎)) ∈ V
5958inex2 5002 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ∈ V
6030, 59eqeltri 2888 . . . . . . . . . . . . . . . . . . . 20 𝑄 ∈ V
6160cnvex 7346 . . . . . . . . . . . . . . . . . . 19 𝑄 ∈ V
6261imaex 7337 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ∈ V
6362f1oen 8216 . . . . . . . . . . . . . . . . 17 ((𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})) → (𝑄 “ {𝑤}) ≈ (𝐽 “ (𝑄 “ {𝑤})))
6446, 57, 633syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽 “ (𝑄 “ {𝑤})))
65 sseqin2 4023 . . . . . . . . . . . . . . . . . . 19 ((𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎) ↔ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤}))
6655, 65mpbi 221 . . . . . . . . . . . . . . . . . 18 ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤})
6766imaeq2i 5681 . . . . . . . . . . . . . . . . 17 (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽 “ (𝑄 “ {𝑤}))
68 isocnv 6807 . . . . . . . . . . . . . . . . . . . 20 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽))
6946, 68syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽))
70 simpr 473 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑤 ∈ (𝑎 × 𝑎))
71 isoini 6815 . . . . . . . . . . . . . . . . . . 19 ((𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽) ∧ 𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
7269, 70, 71syl2anc 575 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
73 fvex 6424 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝑤) ∈ V
7473epini 5712 . . . . . . . . . . . . . . . . . . . 20 ( E “ {(𝐽𝑤)}) = (𝐽𝑤)
7574ineq2i 4017 . . . . . . . . . . . . . . . . . . 19 (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (dom 𝐽 ∩ (𝐽𝑤))
7634oicl 8676 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐽
77 f1of 6356 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7836, 37, 38, 774syl 19 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7978ffvelrnda 6584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ dom 𝐽)
80 ordelss 5959 . . . . . . . . . . . . . . . . . . . . 21 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ⊆ dom 𝐽)
8176, 79, 80sylancr 577 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ⊆ dom 𝐽)
82 sseqin2 4023 . . . . . . . . . . . . . . . . . . . 20 ((𝐽𝑤) ⊆ dom 𝐽 ↔ (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8381, 82sylib 209 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8475, 83syl5eq 2859 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (𝐽𝑤))
8572, 84eqtrd 2847 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽𝑤))
8667, 85syl5eqr 2861 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ (𝑄 “ {𝑤})) = (𝐽𝑤))
8764, 86breqtrd 4877 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽𝑤))
8887ensymd 8246 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≈ (𝑄 “ {𝑤}))
89 infxpen.3 . . . . . . . . . . . . . . . . . . 19 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
90 fvex 6424 . . . . . . . . . . . . . . . . . . . 20 (1st𝑤) ∈ V
91 fvex 6424 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑤) ∈ V
9290, 91unex 7189 . . . . . . . . . . . . . . . . . . 19 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
9389, 92eqeltri 2888 . . . . . . . . . . . . . . . . . 18 𝑀 ∈ V
9493sucex 7244 . . . . . . . . . . . . . . . . 17 suc 𝑀 ∈ V
9594, 94xpex 7195 . . . . . . . . . . . . . . . 16 (suc 𝑀 × suc 𝑀) ∈ V
96 xpss 5334 . . . . . . . . . . . . . . . . . . . 20 (𝑎 × 𝑎) ⊆ (V × V)
97 simp3 1161 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑄 “ {𝑤}))
98 vex 3401 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤 ∈ V
99 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ V
10099eliniseg 5711 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ V → (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤))
10198, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤)
10297, 101sylib 209 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧𝑄𝑤)
10330breqi 4857 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑄𝑤𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤)
104 brin 4903 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
105103, 104bitri 266 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑄𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
106105simprbi 486 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤)
107 brxp 5354 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤 ↔ (𝑧 ∈ (𝑎 × 𝑎) ∧ 𝑤 ∈ (𝑎 × 𝑎)))
108107simplbi 487 . . . . . . . . . . . . . . . . . . . . 21 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤𝑧 ∈ (𝑎 × 𝑎))
109102, 106, 1083syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑎 × 𝑎))
11096, 109sseldi 3803 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (V × V))
11117adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ On)
1121113adant3 1155 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑎 ∈ On)
113 xp1st 7433 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (1st𝑧) ∈ 𝑎)
114 onelon 5968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (1st𝑧) ∈ 𝑎) → (1st𝑧) ∈ On)
115113, 114sylan2 582 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (1st𝑧) ∈ On)
116112, 109, 115syl2anc 575 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ On)
117 eloni 5953 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ On → Ord 𝑎)
118 elxp7 7436 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ (𝑎 × 𝑎) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎)))
119118simprbi 486 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎))
120 ordunel 7260 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝑎 ∧ (1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
1211203expib 1145 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑎 → (((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
122117, 119, 121syl2im 40 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ On → (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
123111, 70, 122sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
12489, 123syl5eqel 2896 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑀𝑎)
125 simprr 780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 𝑚𝑎)
12613, 125sylbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∀𝑚𝑎 𝑚𝑎)
127 simprl 778 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
12813, 127sylbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ω ⊆ 𝑎)
129 iscard 9087 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 ↔ (𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎))
130 cardlim 9084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎))
131 sseq2 3831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (ω ⊆ (card‘𝑎) ↔ ω ⊆ 𝑎))
132 limeq 5955 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (Lim (card‘𝑎) ↔ Lim 𝑎))
133131, 132bibi12d 336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((card‘𝑎) = 𝑎 → ((ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎)) ↔ (ω ⊆ 𝑎 ↔ Lim 𝑎)))
134130, 133mpbii 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 → (ω ⊆ 𝑎 ↔ Lim 𝑎))
135129, 134sylbir 226 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → (ω ⊆ 𝑎 ↔ Lim 𝑎))
136135biimpa 464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) ∧ ω ⊆ 𝑎) → Lim 𝑎)
13717, 126, 128, 136syl21anc 857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → Lim 𝑎)
138137adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → Lim 𝑎)
139 limsuc 7282 . . . . . . . . . . . . . . . . . . . . . . . 24 (Lim 𝑎 → (𝑀𝑎 ↔ suc 𝑀𝑎))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑀𝑎 ↔ suc 𝑀𝑎))
141124, 140mpbid 223 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
142 onelon 5968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ suc 𝑀𝑎) → suc 𝑀 ∈ On)
14317, 141, 142syl2an2r 667 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀 ∈ On)
1441433adant3 1155 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → suc 𝑀 ∈ On)
145 ssun1 3982 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
146145a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
147105simplbi 487 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧𝑅𝑤)
148 df-br 4852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧𝑅𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑅)
14923eleq2i 2884 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑤⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))})
150 opabid 5184 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))} ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
151148, 149, 1503bitri 288 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧𝑅𝑤 ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
152151simprbi 486 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
153 simpl 470 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤) → ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)))
154153orim2i 925 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)) → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
155152, 154syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
156 fvex 6424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1st𝑧) ∈ V
157 fvex 6424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2nd𝑧) ∈ V
158156, 157unex 7189 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
159158elsuc 6013 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
160155, 159sylibr 225 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)))
161 suceq 6009 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 = ((1st𝑤) ∪ (2nd𝑤)) → suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤)))
16289, 161ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤))
163160, 162syl6eleqr 2903 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
164102, 147, 1633syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
165 ontr2 5991 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (1st𝑧) ∈ suc 𝑀))
166165imp 395 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (1st𝑧) ∈ suc 𝑀)
167116, 144, 146, 164, 166syl22anc 858 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ suc 𝑀)
168 xp2nd 7434 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (2nd𝑧) ∈ 𝑎)
169 onelon 5968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (2nd𝑧) ∈ 𝑎) → (2nd𝑧) ∈ On)
170168, 169sylan2 582 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (2nd𝑧) ∈ On)
171112, 109, 170syl2anc 575 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ On)
172 ssun2 3983 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
173172a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
174 ontr2 5991 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (2nd𝑧) ∈ suc 𝑀))
175174imp 395 . . . . . . . . . . . . . . . . . . . 20 ((((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (2nd𝑧) ∈ suc 𝑀)
176171, 144, 173, 164, 175syl22anc 858 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ suc 𝑀)
177 elxp7 7436 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (suc 𝑀 × suc 𝑀) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)))
178177biimpri 219 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
179110, 167, 176, 178syl12anc 856 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
1801793expia 1143 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑧 ∈ (𝑄 “ {𝑤}) → 𝑧 ∈ (suc 𝑀 × suc 𝑀)))
181180ssrdv 3811 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀))
182 ssdomg 8241 . . . . . . . . . . . . . . . 16 ((suc 𝑀 × suc 𝑀) ∈ V → ((𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀)))
18395, 181, 182mpsyl 68 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀))
184128adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ω ⊆ 𝑎)
185 nnfi 8395 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
186 xpfi 8473 . . . . . . . . . . . . . . . . . . . . . 22 ((suc 𝑀 ∈ Fin ∧ suc 𝑀 ∈ Fin) → (suc 𝑀 × suc 𝑀) ∈ Fin)
187186anidms 558 . . . . . . . . . . . . . . . . . . . . 21 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ∈ Fin)
188 isfinite 8799 . . . . . . . . . . . . . . . . . . . . 21 ((suc 𝑀 × suc 𝑀) ∈ Fin ↔ (suc 𝑀 × suc 𝑀) ≺ ω)
189187, 188sylib 209 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ≺ ω)
190185, 189syl 17 . . . . . . . . . . . . . . . . . . 19 (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ ω)
191 ssdomg 8241 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (ω ⊆ 𝑎 → ω ≼ 𝑎))
19214, 191ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (ω ⊆ 𝑎 → ω ≼ 𝑎)
193 sdomdomtr 8335 . . . . . . . . . . . . . . . . . . 19 (((suc 𝑀 × suc 𝑀) ≺ ω ∧ ω ≼ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
194190, 192, 193syl2an 585 . . . . . . . . . . . . . . . . . 18 ((suc 𝑀 ∈ ω ∧ ω ⊆ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
195194expcom 400 . . . . . . . . . . . . . . . . 17 (ω ⊆ 𝑎 → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
196184, 195syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
197 breq1 4854 . . . . . . . . . . . . . . . . . 18 (𝑚 = suc 𝑀 → (𝑚𝑎 ↔ suc 𝑀𝑎))
198126adantr 468 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 𝑚𝑎)
199197, 198, 141rspcdva 3515 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
200 omelon 8793 . . . . . . . . . . . . . . . . . . 19 ω ∈ On
201 ontri1 5977 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ suc 𝑀 ∈ On) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
202200, 143, 201sylancr 577 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
203 sseq2 3831 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → (ω ⊆ 𝑚 ↔ ω ⊆ suc 𝑀))
204 xpeq12 5342 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = suc 𝑀𝑚 = suc 𝑀) → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
205204anidms 558 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀 → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
206 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀𝑚 = suc 𝑀)
207205, 206breq12d 4864 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → ((𝑚 × 𝑚) ≈ 𝑚 ↔ (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
208203, 207imbi12d 335 . . . . . . . . . . . . . . . . . . 19 (𝑚 = suc 𝑀 → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ↔ (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀)))
209 simplr 776 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
21013, 209sylbi 208 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
211210adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
212208, 211, 141rspcdva 3515 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
213202, 212sylbird 251 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
214 ensdomtr 8338 . . . . . . . . . . . . . . . . . 18 (((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 ∧ suc 𝑀𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
215214expcom 400 . . . . . . . . . . . . . . . . 17 (suc 𝑀𝑎 → ((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
216199, 213, 215sylsyld 61 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
217196, 216pm2.61d 171 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
218 domsdomtr 8337 . . . . . . . . . . . . . . 15 (((𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀) ∧ (suc 𝑀 × suc 𝑀) ≺ 𝑎) → (𝑄 “ {𝑤}) ≺ 𝑎)
219183, 217, 218syl2anc 575 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≺ 𝑎)
220 ensdomtr 8338 . . . . . . . . . . . . . 14 (((𝐽𝑤) ≈ (𝑄 “ {𝑤}) ∧ (𝑄 “ {𝑤}) ≺ 𝑎) → (𝐽𝑤) ≺ 𝑎)
22188, 219, 220syl2anc 575 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≺ 𝑎)
222 ordelon 5967 . . . . . . . . . . . . . . 15 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ∈ On)
22376, 79, 222sylancr 577 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ On)
224 onenon 9061 . . . . . . . . . . . . . . 15 (𝑎 ∈ On → 𝑎 ∈ dom card)
225111, 224syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ dom card)
226 cardsdomel 9086 . . . . . . . . . . . . . 14 (((𝐽𝑤) ∈ On ∧ 𝑎 ∈ dom card) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
227223, 225, 226syl2anc 575 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
228221, 227mpbid 223 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ (card‘𝑎))
229 eleq2 2881 . . . . . . . . . . . . . 14 ((card‘𝑎) = 𝑎 → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
230129, 229sylbir 226 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
23117, 198, 230syl2an2r 667 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
232228, 231mpbid 223 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ 𝑎)
233232ralrimiva 3161 . . . . . . . . . 10 (𝜑 → ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎)
234 fnfvrnss 6615 . . . . . . . . . . 11 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
235 ssdomg 8241 . . . . . . . . . . 11 (𝑎 ∈ V → (ran 𝐽𝑎 → ran 𝐽𝑎))
23614, 234, 235mpsyl 68 . . . . . . . . . 10 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
23745, 233, 236syl2anc 575 . . . . . . . . 9 (𝜑 → ran 𝐽𝑎)
238 endomtr 8253 . . . . . . . . 9 (((𝑎 × 𝑎) ≈ ran 𝐽 ∧ ran 𝐽𝑎) → (𝑎 × 𝑎) ≼ 𝑎)
23943, 237, 238syl2anc 575 . . . . . . . 8 (𝜑 → (𝑎 × 𝑎) ≼ 𝑎)
24013, 239sylbir 226 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≼ 𝑎)
241 df1o2 7812 . . . . . . . . . . . 12 1𝑜 = {∅}
242 1onn 7959 . . . . . . . . . . . 12 1𝑜 ∈ ω
243241, 242eqeltrri 2889 . . . . . . . . . . 11 {∅} ∈ ω
244 nnsdom 8801 . . . . . . . . . . 11 ({∅} ∈ ω → {∅} ≺ ω)
245 sdomdom 8223 . . . . . . . . . . 11 ({∅} ≺ ω → {∅} ≼ ω)
246243, 244, 245mp2b 10 . . . . . . . . . 10 {∅} ≼ ω
247 domtr 8248 . . . . . . . . . 10 (({∅} ≼ ω ∧ ω ≼ 𝑎) → {∅} ≼ 𝑎)
248246, 192, 247sylancr 577 . . . . . . . . 9 (ω ⊆ 𝑎 → {∅} ≼ 𝑎)
249 0ex 4991 . . . . . . . . . . . 12 ∅ ∈ V
25014, 249xpsnen 8286 . . . . . . . . . . 11 (𝑎 × {∅}) ≈ 𝑎
251250ensymi 8245 . . . . . . . . . 10 𝑎 ≈ (𝑎 × {∅})
25214xpdom2 8297 . . . . . . . . . 10 ({∅} ≼ 𝑎 → (𝑎 × {∅}) ≼ (𝑎 × 𝑎))
253 endomtr 8253 . . . . . . . . . 10 ((𝑎 ≈ (𝑎 × {∅}) ∧ (𝑎 × {∅}) ≼ (𝑎 × 𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
254251, 252, 253sylancr 577 . . . . . . . . 9 ({∅} ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎))
255248, 254syl 17 . . . . . . . 8 (ω ⊆ 𝑎𝑎 ≼ (𝑎 × 𝑎))
256255ad2antrl 710 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
257 sbth 8322 . . . . . . 7 (((𝑎 × 𝑎) ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
258240, 256, 257syl2anc 575 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
259258expr 446 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
260 simplr 776 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
261 simpll 774 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
262 simprr 780 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ¬ ∀𝑚𝑎 𝑚𝑎)
263 rexnal 3189 . . . . . . . . . 10 (∃𝑚𝑎 ¬ 𝑚𝑎 ↔ ¬ ∀𝑚𝑎 𝑚𝑎)
264 onelss 5985 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
265 ssdomg 8241 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
266264, 265syld 47 . . . . . . . . . . . 12 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
267 bren2 8226 . . . . . . . . . . . . 13 (𝑚𝑎 ↔ (𝑚𝑎 ∧ ¬ 𝑚𝑎))
268267simplbi2 490 . . . . . . . . . . . 12 (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎))
269266, 268syl6 35 . . . . . . . . . . 11 (𝑎 ∈ On → (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎)))
270269reximdvai 3209 . . . . . . . . . 10 (𝑎 ∈ On → (∃𝑚𝑎 ¬ 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
271263, 270syl5bir 234 . . . . . . . . 9 (𝑎 ∈ On → (¬ ∀𝑚𝑎 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
272261, 262, 271sylc 65 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 𝑚𝑎)
273 r19.29 3267 . . . . . . . 8 ((∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ ∃𝑚𝑎 𝑚𝑎) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
274260, 272, 273syl2anc 575 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
275 simprl 778 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
276 onelon 5968 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ 𝑚𝑎) → 𝑚 ∈ On)
277 ensym 8244 . . . . . . . . . . . . . . . . . 18 (𝑚𝑎𝑎𝑚)
278 domentr 8254 . . . . . . . . . . . . . . . . . 18 ((ω ≼ 𝑎𝑎𝑚) → ω ≼ 𝑚)
279192, 277, 278syl2an 585 . . . . . . . . . . . . . . . . 17 ((ω ⊆ 𝑎𝑚𝑎) → ω ≼ 𝑚)
280 domnsym 8328 . . . . . . . . . . . . . . . . . . 19 (ω ≼ 𝑚 → ¬ 𝑚 ≺ ω)
281 nnsdom 8801 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ω → 𝑚 ≺ ω)
282280, 281nsyl 137 . . . . . . . . . . . . . . . . . 18 (ω ≼ 𝑚 → ¬ 𝑚 ∈ ω)
283 ontri1 5977 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ 𝑚 ∈ On) → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
284200, 283mpan 673 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
285282, 284syl5ibr 237 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ On → (ω ≼ 𝑚 → ω ⊆ 𝑚))
286276, 279, 285syl2im 40 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑚𝑎) → ((ω ⊆ 𝑎𝑚𝑎) → ω ⊆ 𝑚))
287286expd 402 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑚𝑎) → (ω ⊆ 𝑎 → (𝑚𝑎 → ω ⊆ 𝑚)))
288287impcom 396 . . . . . . . . . . . . . 14 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (𝑚𝑎 → ω ⊆ 𝑚))
289288imim1d 82 . . . . . . . . . . . . 13 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (𝑚𝑎 → (𝑚 × 𝑚) ≈ 𝑚)))
290289imp32 407 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑚 × 𝑚) ≈ 𝑚)
291 entr 8247 . . . . . . . . . . . . . . . 16 (((𝑚 × 𝑚) ≈ 𝑚𝑚𝑎) → (𝑚 × 𝑚) ≈ 𝑎)
292291ancoms 448 . . . . . . . . . . . . . . 15 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑚 × 𝑚) ≈ 𝑎)
293 xpen 8365 . . . . . . . . . . . . . . . . 17 ((𝑎𝑚𝑎𝑚) → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
294293anidms 558 . . . . . . . . . . . . . . . 16 (𝑎𝑚 → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
295 entr 8247 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑎) ≈ (𝑚 × 𝑚) ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
296294, 295sylan 571 . . . . . . . . . . . . . . 15 ((𝑎𝑚 ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
297277, 292, 296syl2an2r 667 . . . . . . . . . . . . . 14 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑎 × 𝑎) ≈ 𝑎)
298297ex 399 . . . . . . . . . . . . 13 (𝑚𝑎 → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
299298ad2antll 711 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
300290, 299mpd 15 . . . . . . . . . . 11 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
301300ex 399 . . . . . . . . . 10 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
302301expr 446 . . . . . . . . 9 ((ω ⊆ 𝑎𝑎 ∈ On) → (𝑚𝑎 → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎)))
303302rexlimdv 3225 . . . . . . . 8 ((ω ⊆ 𝑎𝑎 ∈ On) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
304275, 261, 303syl2anc 575 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
305274, 304mpd 15 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
306305expr 446 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (¬ ∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
307259, 306pm2.61d 171 . . . 4 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
308307exp31 408 . . 3 (𝑎 ∈ On → (∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎)))
3096, 12, 308tfis3 7290 . 2 (𝐴 ∈ On → (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴))
310309imp 395 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2157  wral 3103  wrex 3104  Vcvv 3398  cun 3774  cin 3775  wss 3776  c0 4123  {csn 4377  cop 4383   class class class wbr 4851  {copab 4913   E cep 5230   Se wse 5275   We wwe 5276   × cxp 5316  ccnv 5317  dom cdm 5318  ran crn 5319  cres 5320  cima 5321  Ord word 5942  Oncon0 5943  Lim wlim 5944  suc csuc 5945   Fn wfn 6099  wf 6100  1-1wf1 6101  1-1-ontowf1o 6103  cfv 6104   Isom wiso 6105  ωcom 7298  1st c1st 7399  2nd c2nd 7400  1𝑜c1o 7792  cen 8192  cdom 8193  csdm 8194  Fincfn 8195  OrdIsocoi 8656  cardccrd 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-card 9051
This theorem is referenced by:  infxpen  9123
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