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Theorem infxpenlem 10049
Description: Lemma for infxpen 10050. (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 4005 . . . 4 (𝑎 = 𝑚 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝑚))
2 xpeq12 5699 . . . . . 6 ((𝑎 = 𝑚𝑎 = 𝑚) → (𝑎 × 𝑎) = (𝑚 × 𝑚))
32anidms 565 . . . . 5 (𝑎 = 𝑚 → (𝑎 × 𝑎) = (𝑚 × 𝑚))
4 id 22 . . . . 5 (𝑎 = 𝑚𝑎 = 𝑚)
53, 4breq12d 5158 . . . 4 (𝑎 = 𝑚 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝑚 × 𝑚) ≈ 𝑚))
61, 5imbi12d 343 . . 3 (𝑎 = 𝑚 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)))
7 sseq2 4005 . . . 4 (𝑎 = 𝐴 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝐴))
8 xpeq12 5699 . . . . . 6 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑎 × 𝑎) = (𝐴 × 𝐴))
98anidms 565 . . . . 5 (𝑎 = 𝐴 → (𝑎 × 𝑎) = (𝐴 × 𝐴))
10 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
119, 10breq12d 5158 . . . 4 (𝑎 = 𝐴 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝐴 × 𝐴) ≈ 𝐴))
127, 11imbi12d 343 . . 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 3466 . . . . . . . . . . . . 13 𝑎 ∈ V
1514, 14xpex 7753 . . . . . . . . . . . 12 (𝑎 × 𝑎) ∈ V
16 simpll 765 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
1713, 16sylbi 216 . . . . . . . . . . . . . . . . 17 (𝜑𝑎 ∈ On)
18 onss 7785 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ On → 𝑎 ⊆ On)
1917, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑎 ⊆ On)
20 xpss12 5689 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ On ∧ 𝑎 ⊆ On) → (𝑎 × 𝑎) ⊆ (On × On))
2119, 19, 20syl2anc 582 . . . . . . . . . . . . . . 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 10048 . . . . . . . . . . . . . . . 16 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
2524simpli 482 . . . . . . . . . . . . . . 15 𝑅 We (On × On)
26 wess 5661 . . . . . . . . . . . . . . 15 ((𝑎 × 𝑎) ⊆ (On × On) → (𝑅 We (On × On) → 𝑅 We (𝑎 × 𝑎)))
2721, 25, 26mpisyl 21 . . . . . . . . . . . . . 14 (𝜑𝑅 We (𝑎 × 𝑎))
28 weinxp 5758 . . . . . . . . . . . . . 14 (𝑅 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
2927, 28sylib 217 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
30 infxpen.1 . . . . . . . . . . . . . 14 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
31 weeq1 5662 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . 12 (𝜑𝑄 We (𝑎 × 𝑎))
34 infxpen.4 . . . . . . . . . . . . 13 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
3534oiiso 9573 . . . . . . . . . . . 12 (((𝑎 × 𝑎) ∈ V ∧ 𝑄 We (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
3615, 33, 35sylancr 585 . . . . . . . . . . 11 (𝜑𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
37 isof1o 7327 . . . . . . . . . . 11 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎))
38 f1ocnv 6847 . . . . . . . . . . 11 (𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎) → 𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽)
39 f1of1 6834 . . . . . . . . . . 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 6846 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽)
4215f1oen 8996 . . . . . . . . . 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 6836 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽 Fn (𝑎 × 𝑎))
4536, 37, 38, 444syl 19 . . . . . . . . . 10 (𝜑𝐽 Fn (𝑎 × 𝑎))
4636adantr 479 . . . . . . . . . . . . . . . . 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 6083 . . . . . . . . . . . . . . . . . . 19 (𝑄 “ {𝑤}) ⊆ dom 𝑄
49 inss2 4228 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
5030, 49eqsstri 4013 . . . . . . . . . . . . . . . . . . . . 21 𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
51 dmss 5901 . . . . . . . . . . . . . . . . . . . . 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 5928 . . . . . . . . . . . . . . . . . . . 20 dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)) = (𝑎 × 𝑎)
5452, 53sseqtri 4015 . . . . . . . . . . . . . . . . . . 19 dom 𝑄 ⊆ (𝑎 × 𝑎)
5548, 54sstri 3988 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)
56 f1ores 6849 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 17 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5815, 15xpex 7753 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 × 𝑎) × (𝑎 × 𝑎)) ∈ V
5958inex2 5315 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ∈ V
6030, 59eqeltri 2822 . . . . . . . . . . . . . . . . . . . 20 𝑄 ∈ V
6160cnvex 7930 . . . . . . . . . . . . . . . . . . 19 𝑄 ∈ V
6261imaex 7919 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ∈ V
6362f1oen 8996 . . . . . . . . . . . . . . . . 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 4213 . . . . . . . . . . . . . . . . . . 19 ((𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎) ↔ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤}))
6655, 65mpbi 229 . . . . . . . . . . . . . . . . . 18 ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤})
6766imaeq2i 6059 . . . . . . . . . . . . . . . . 17 (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽 “ (𝑄 “ {𝑤}))
68 isocnv 7334 . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑤 ∈ (𝑎 × 𝑎))
71 isoini 7342 . . . . . . . . . . . . . . . . . . 19 ((𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽) ∧ 𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
7269, 70, 71syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
73 fvex 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝑤) ∈ V
7473epini 6098 . . . . . . . . . . . . . . . . . . . 20 ( E “ {(𝐽𝑤)}) = (𝐽𝑤)
7574ineq2i 4207 . . . . . . . . . . . . . . . . . . 19 (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (dom 𝐽 ∩ (𝐽𝑤))
7634oicl 9565 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐽
77 f1of 6835 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7836, 37, 38, 774syl 19 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7978ffvelcdmda 7090 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ dom 𝐽)
80 ordelss 6384 . . . . . . . . . . . . . . . . . . . . 21 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ⊆ dom 𝐽)
8176, 79, 80sylancr 585 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ⊆ dom 𝐽)
82 sseqin2 4213 . . . . . . . . . . . . . . . . . . . 20 ((𝐽𝑤) ⊆ dom 𝐽 ↔ (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8381, 82sylib 217 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8475, 83eqtrid 2778 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (𝐽𝑤))
8572, 84eqtrd 2766 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽𝑤))
8667, 85eqtr3id 2780 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ (𝑄 “ {𝑤})) = (𝐽𝑤))
8764, 86breqtrd 5171 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽𝑤))
8887ensymd 9028 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≈ (𝑄 “ {𝑤}))
89 infxpen.3 . . . . . . . . . . . . . . . . . . 19 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
90 fvex 6906 . . . . . . . . . . . . . . . . . . . 20 (1st𝑤) ∈ V
91 fvex 6906 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑤) ∈ V
9290, 91unex 7746 . . . . . . . . . . . . . . . . . . 19 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
9389, 92eqeltri 2822 . . . . . . . . . . . . . . . . . 18 𝑀 ∈ V
9493sucex 7807 . . . . . . . . . . . . . . . . 17 suc 𝑀 ∈ V
9594, 94xpex 7753 . . . . . . . . . . . . . . . 16 (suc 𝑀 × suc 𝑀) ∈ V
96 xpss 5690 . . . . . . . . . . . . . . . . . . . 20 (𝑎 × 𝑎) ⊆ (V × V)
97 simp3 1135 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑄 “ {𝑤}))
98 vex 3466 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ V
9998eliniseg 6096 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ V → (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤))
10099elv 3468 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤)
10197, 100sylib 217 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧𝑄𝑤)
10230breqi 5151 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑄𝑤𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤)
103 brin 5197 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
104102, 103bitri 274 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑄𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
105104simprbi 495 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤)
106 brxp 5723 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤 ↔ (𝑧 ∈ (𝑎 × 𝑎) ∧ 𝑤 ∈ (𝑎 × 𝑎)))
107106simplbi 496 . . . . . . . . . . . . . . . . . . . . 21 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤𝑧 ∈ (𝑎 × 𝑎))
108101, 105, 1073syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑎 × 𝑎))
10996, 108sselid 3976 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (V × V))
11017adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ On)
1111103adant3 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑎 ∈ On)
112 xp1st 8027 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (1st𝑧) ∈ 𝑎)
113 onelon 6393 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (1st𝑧) ∈ 𝑎) → (1st𝑧) ∈ On)
114112, 113sylan2 591 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (1st𝑧) ∈ On)
115111, 108, 114syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ On)
116 eloni 6378 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ On → Ord 𝑎)
117 elxp7 8030 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ (𝑎 × 𝑎) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎)))
118117simprbi 495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎))
119 ordunel 7828 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝑎 ∧ (1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
1201193expib 1119 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑎 → (((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
121116, 118, 120syl2im 40 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ On → (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
122110, 70, 121sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
12389, 122eqeltrid 2830 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑀𝑎)
124 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 𝑚𝑎)
12513, 124sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∀𝑚𝑎 𝑚𝑎)
126 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
12713, 126sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ω ⊆ 𝑎)
128 iscard 10011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 ↔ (𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎))
129 cardlim 10008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎))
130 sseq2 4005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (ω ⊆ (card‘𝑎) ↔ ω ⊆ 𝑎))
131 limeq 6380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (Lim (card‘𝑎) ↔ Lim 𝑎))
132130, 131bibi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((card‘𝑎) = 𝑎 → ((ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎)) ↔ (ω ⊆ 𝑎 ↔ Lim 𝑎)))
133129, 132mpbii 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 → (ω ⊆ 𝑎 ↔ Lim 𝑎))
134128, 133sylbir 234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → (ω ⊆ 𝑎 ↔ Lim 𝑎))
135134biimpa 475 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) ∧ ω ⊆ 𝑎) → Lim 𝑎)
13617, 125, 127, 135syl21anc 836 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → Lim 𝑎)
137136adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → Lim 𝑎)
138 limsuc 7851 . . . . . . . . . . . . . . . . . . . . . . . 24 (Lim 𝑎 → (𝑀𝑎 ↔ suc 𝑀𝑎))
139137, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑀𝑎 ↔ suc 𝑀𝑎))
140123, 139mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
141 onelon 6393 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ suc 𝑀𝑎) → suc 𝑀 ∈ On)
14217, 140, 141syl2an2r 683 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀 ∈ On)
1431423adant3 1129 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → suc 𝑀 ∈ On)
144 ssun1 4170 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
145144a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
146104simplbi 496 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧𝑅𝑤)
147 df-br 5146 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧𝑅𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑅)
14823eleq2i 2818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑤⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))})
149 opabidw 5522 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
150147, 148, 1493bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧𝑅𝑤 ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
151150simprbi 495 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
152 simpl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤) → ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)))
153152orim2i 908 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)) → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
154151, 153syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
155 fvex 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1st𝑧) ∈ V
156 fvex 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2nd𝑧) ∈ V
157155, 156unex 7746 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
158157elsuc 6438 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
159154, 158sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)))
160 suceq 6434 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 = ((1st𝑤) ∪ (2nd𝑤)) → suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤)))
16189, 160ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤))
162159, 161eleqtrrdi 2837 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
163101, 146, 1623syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
164 ontr2 6415 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (1st𝑧) ∈ suc 𝑀))
165164imp 405 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (1st𝑧) ∈ suc 𝑀)
166115, 143, 145, 163, 165syl22anc 837 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ suc 𝑀)
167 xp2nd 8028 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (2nd𝑧) ∈ 𝑎)
168 onelon 6393 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (2nd𝑧) ∈ 𝑎) → (2nd𝑧) ∈ On)
169167, 168sylan2 591 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (2nd𝑧) ∈ On)
170111, 108, 169syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ On)
171 ssun2 4171 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
172171a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
173 ontr2 6415 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (2nd𝑧) ∈ suc 𝑀))
174173imp 405 . . . . . . . . . . . . . . . . . . . 20 ((((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (2nd𝑧) ∈ suc 𝑀)
175170, 143, 172, 163, 174syl22anc 837 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ suc 𝑀)
176 elxp7 8030 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (suc 𝑀 × suc 𝑀) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)))
177176biimpri 227 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
178109, 166, 175, 177syl12anc 835 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
1791783expia 1118 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑧 ∈ (𝑄 “ {𝑤}) → 𝑧 ∈ (suc 𝑀 × suc 𝑀)))
180179ssrdv 3984 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀))
181 ssdomg 9023 . . . . . . . . . . . . . . . 16 ((suc 𝑀 × suc 𝑀) ∈ V → ((𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀)))
18295, 180, 181mpsyl 68 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀))
183127adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ω ⊆ 𝑎)
184 nnfi 9197 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
185 xpfi 9353 . . . . . . . . . . . . . . . . . . . . . 22 ((suc 𝑀 ∈ Fin ∧ suc 𝑀 ∈ Fin) → (suc 𝑀 × suc 𝑀) ∈ Fin)
186185anidms 565 . . . . . . . . . . . . . . . . . . . . 21 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ∈ Fin)
187 isfinite 9688 . . . . . . . . . . . . . . . . . . . . 21 ((suc 𝑀 × suc 𝑀) ∈ Fin ↔ (suc 𝑀 × suc 𝑀) ≺ ω)
188186, 187sylib 217 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ≺ ω)
189184, 188syl 17 . . . . . . . . . . . . . . . . . . 19 (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ ω)
190 ssdomg 9023 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (ω ⊆ 𝑎 → ω ≼ 𝑎))
191190elv 3468 . . . . . . . . . . . . . . . . . . 19 (ω ⊆ 𝑎 → ω ≼ 𝑎)
192 sdomdomtr 9140 . . . . . . . . . . . . . . . . . . 19 (((suc 𝑀 × suc 𝑀) ≺ ω ∧ ω ≼ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
193189, 191, 192syl2an 594 . . . . . . . . . . . . . . . . . 18 ((suc 𝑀 ∈ ω ∧ ω ⊆ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
194193expcom 412 . . . . . . . . . . . . . . . . 17 (ω ⊆ 𝑎 → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
195183, 194syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
196 breq1 5148 . . . . . . . . . . . . . . . . . 18 (𝑚 = suc 𝑀 → (𝑚𝑎 ↔ suc 𝑀𝑎))
197125adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 𝑚𝑎)
198196, 197, 140rspcdva 3608 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
199 omelon 9682 . . . . . . . . . . . . . . . . . . 19 ω ∈ On
200 ontri1 6402 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ suc 𝑀 ∈ On) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
201199, 142, 200sylancr 585 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
202 sseq2 4005 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → (ω ⊆ 𝑚 ↔ ω ⊆ suc 𝑀))
203 xpeq12 5699 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = suc 𝑀𝑚 = suc 𝑀) → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
204203anidms 565 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀 → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
205 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀𝑚 = suc 𝑀)
206204, 205breq12d 5158 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → ((𝑚 × 𝑚) ≈ 𝑚 ↔ (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
207202, 206imbi12d 343 . . . . . . . . . . . . . . . . . . 19 (𝑚 = suc 𝑀 → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ↔ (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀)))
208 simplr 767 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
20913, 208sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
210209adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
211207, 210, 140rspcdva 3608 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
212201, 211sylbird 259 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
213 ensdomtr 9143 . . . . . . . . . . . . . . . . . 18 (((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 ∧ suc 𝑀𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
214213expcom 412 . . . . . . . . . . . . . . . . 17 (suc 𝑀𝑎 → ((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
215198, 212, 214sylsyld 61 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
216195, 215pm2.61d 179 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
217 domsdomtr 9142 . . . . . . . . . . . . . . 15 (((𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀) ∧ (suc 𝑀 × suc 𝑀) ≺ 𝑎) → (𝑄 “ {𝑤}) ≺ 𝑎)
218182, 216, 217syl2anc 582 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≺ 𝑎)
219 ensdomtr 9143 . . . . . . . . . . . . . 14 (((𝐽𝑤) ≈ (𝑄 “ {𝑤}) ∧ (𝑄 “ {𝑤}) ≺ 𝑎) → (𝐽𝑤) ≺ 𝑎)
22088, 218, 219syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≺ 𝑎)
221 ordelon 6392 . . . . . . . . . . . . . . 15 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ∈ On)
22276, 79, 221sylancr 585 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ On)
223 onenon 9985 . . . . . . . . . . . . . . 15 (𝑎 ∈ On → 𝑎 ∈ dom card)
224110, 223syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ dom card)
225 cardsdomel 10010 . . . . . . . . . . . . . 14 (((𝐽𝑤) ∈ On ∧ 𝑎 ∈ dom card) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
226222, 224, 225syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
227220, 226mpbid 231 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ (card‘𝑎))
228 eleq2 2815 . . . . . . . . . . . . . 14 ((card‘𝑎) = 𝑎 → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
229128, 228sylbir 234 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
23017, 197, 229syl2an2r 683 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
231227, 230mpbid 231 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ 𝑎)
232231ralrimiva 3136 . . . . . . . . . 10 (𝜑 → ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎)
233 fnfvrnss 7127 . . . . . . . . . . 11 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
234 ssdomg 9023 . . . . . . . . . . 11 (𝑎 ∈ V → (ran 𝐽𝑎 → ran 𝐽𝑎))
23514, 233, 234mpsyl 68 . . . . . . . . . 10 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
23645, 232, 235syl2anc 582 . . . . . . . . 9 (𝜑 → ran 𝐽𝑎)
237 endomtr 9035 . . . . . . . . 9 (((𝑎 × 𝑎) ≈ ran 𝐽 ∧ ran 𝐽𝑎) → (𝑎 × 𝑎) ≼ 𝑎)
23843, 236, 237syl2anc 582 . . . . . . . 8 (𝜑 → (𝑎 × 𝑎) ≼ 𝑎)
23913, 238sylbir 234 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≼ 𝑎)
240 df1o2 8495 . . . . . . . . . . . 12 1o = {∅}
241 1onn 8662 . . . . . . . . . . . 12 1o ∈ ω
242240, 241eqeltrri 2823 . . . . . . . . . . 11 {∅} ∈ ω
243 nnsdom 9690 . . . . . . . . . . 11 ({∅} ∈ ω → {∅} ≺ ω)
244 sdomdom 9003 . . . . . . . . . . 11 ({∅} ≺ ω → {∅} ≼ ω)
245242, 243, 244mp2b 10 . . . . . . . . . 10 {∅} ≼ ω
246 domtr 9030 . . . . . . . . . 10 (({∅} ≼ ω ∧ ω ≼ 𝑎) → {∅} ≼ 𝑎)
247245, 191, 246sylancr 585 . . . . . . . . 9 (ω ⊆ 𝑎 → {∅} ≼ 𝑎)
248 0ex 5304 . . . . . . . . . . . 12 ∅ ∈ V
24914, 248xpsnen 9085 . . . . . . . . . . 11 (𝑎 × {∅}) ≈ 𝑎
250249ensymi 9027 . . . . . . . . . 10 𝑎 ≈ (𝑎 × {∅})
25114xpdom2 9097 . . . . . . . . . 10 ({∅} ≼ 𝑎 → (𝑎 × {∅}) ≼ (𝑎 × 𝑎))
252 endomtr 9035 . . . . . . . . . 10 ((𝑎 ≈ (𝑎 × {∅}) ∧ (𝑎 × {∅}) ≼ (𝑎 × 𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
253250, 251, 252sylancr 585 . . . . . . . . 9 ({∅} ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎))
254247, 253syl 17 . . . . . . . 8 (ω ⊆ 𝑎𝑎 ≼ (𝑎 × 𝑎))
255254ad2antrl 726 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
256 sbth 9123 . . . . . . 7 (((𝑎 × 𝑎) ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
257239, 255, 256syl2anc 582 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
258257expr 455 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
259 simplr 767 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
260 simpll 765 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
261 simprr 771 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ¬ ∀𝑚𝑎 𝑚𝑎)
262 rexnal 3090 . . . . . . . . . 10 (∃𝑚𝑎 ¬ 𝑚𝑎 ↔ ¬ ∀𝑚𝑎 𝑚𝑎)
263 onelss 6410 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
264 ssdomg 9023 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
265263, 264syld 47 . . . . . . . . . . . 12 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
266 bren2 9006 . . . . . . . . . . . . 13 (𝑚𝑎 ↔ (𝑚𝑎 ∧ ¬ 𝑚𝑎))
267266simplbi2 499 . . . . . . . . . . . 12 (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎))
268265, 267syl6 35 . . . . . . . . . . 11 (𝑎 ∈ On → (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎)))
269268reximdvai 3155 . . . . . . . . . 10 (𝑎 ∈ On → (∃𝑚𝑎 ¬ 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
270262, 269biimtrrid 242 . . . . . . . . 9 (𝑎 ∈ On → (¬ ∀𝑚𝑎 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
271260, 261, 270sylc 65 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 𝑚𝑎)
272 r19.29 3104 . . . . . . . 8 ((∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ ∃𝑚𝑎 𝑚𝑎) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
273259, 271, 272syl2anc 582 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
274 simprl 769 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
275 onelon 6393 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ 𝑚𝑎) → 𝑚 ∈ On)
276 ensym 9026 . . . . . . . . . . . . . . . . . 18 (𝑚𝑎𝑎𝑚)
277 domentr 9036 . . . . . . . . . . . . . . . . . 18 ((ω ≼ 𝑎𝑎𝑚) → ω ≼ 𝑚)
278191, 276, 277syl2an 594 . . . . . . . . . . . . . . . . 17 ((ω ⊆ 𝑎𝑚𝑎) → ω ≼ 𝑚)
279 domnsym 9129 . . . . . . . . . . . . . . . . . . 19 (ω ≼ 𝑚 → ¬ 𝑚 ≺ ω)
280 nnsdom 9690 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ω → 𝑚 ≺ ω)
281279, 280nsyl 140 . . . . . . . . . . . . . . . . . 18 (ω ≼ 𝑚 → ¬ 𝑚 ∈ ω)
282 ontri1 6402 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ 𝑚 ∈ On) → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
283199, 282mpan 688 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
284281, 283imbitrrid 245 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ On → (ω ≼ 𝑚 → ω ⊆ 𝑚))
285275, 278, 284syl2im 40 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑚𝑎) → ((ω ⊆ 𝑎𝑚𝑎) → ω ⊆ 𝑚))
286285expd 414 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑚𝑎) → (ω ⊆ 𝑎 → (𝑚𝑎 → ω ⊆ 𝑚)))
287286impcom 406 . . . . . . . . . . . . . 14 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (𝑚𝑎 → ω ⊆ 𝑚))
288287imim1d 82 . . . . . . . . . . . . 13 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (𝑚𝑎 → (𝑚 × 𝑚) ≈ 𝑚)))
289288imp32 417 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑚 × 𝑚) ≈ 𝑚)
290 entr 9029 . . . . . . . . . . . . . . . 16 (((𝑚 × 𝑚) ≈ 𝑚𝑚𝑎) → (𝑚 × 𝑚) ≈ 𝑎)
291290ancoms 457 . . . . . . . . . . . . . . 15 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑚 × 𝑚) ≈ 𝑎)
292 xpen 9170 . . . . . . . . . . . . . . . . 17 ((𝑎𝑚𝑎𝑚) → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
293292anidms 565 . . . . . . . . . . . . . . . 16 (𝑎𝑚 → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
294 entr 9029 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑎) ≈ (𝑚 × 𝑚) ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
295293, 294sylan 578 . . . . . . . . . . . . . . 15 ((𝑎𝑚 ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
296276, 291, 295syl2an2r 683 . . . . . . . . . . . . . 14 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑎 × 𝑎) ≈ 𝑎)
297296ex 411 . . . . . . . . . . . . 13 (𝑚𝑎 → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
298297ad2antll 727 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
299289, 298mpd 15 . . . . . . . . . . 11 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
300299ex 411 . . . . . . . . . 10 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
301300expr 455 . . . . . . . . 9 ((ω ⊆ 𝑎𝑎 ∈ On) → (𝑚𝑎 → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎)))
302301rexlimdv 3143 . . . . . . . 8 ((ω ⊆ 𝑎𝑎 ∈ On) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
303274, 260, 302syl2anc 582 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
304273, 303mpd 15 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
305304expr 455 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (¬ ∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
306258, 305pm2.61d 179 . . . 4 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
307306exp31 418 . . 3 (𝑎 ∈ On → (∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎)))
3086, 12, 307tfis3 7860 . 2 (𝐴 ∈ On → (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴))
309308imp 405 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  Vcvv 3462  cun 3944  cin 3945  wss 3946  c0 4322  {csn 4623  cop 4629   class class class wbr 5145  {copab 5207   E cep 5577   Se wse 5627   We wwe 5628   × cxp 5672  ccnv 5673  dom cdm 5674  ran crn 5675  cres 5676  cima 5677  Ord word 6367  Oncon0 6368  Lim wlim 6369  suc csuc 6370   Fn wfn 6541  wf 6542  1-1wf1 6543  1-1-ontowf1o 6545  cfv 6546   Isom wiso 6547  ωcom 7868  1st c1st 7993  2nd c2nd 7994  1oc1o 8481  cen 8963  cdom 8964  csdm 8965  Fincfn 8966  OrdIsocoi 9545  cardccrd 9971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-inf2 9677
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-oi 9546  df-card 9975
This theorem is referenced by:  infxpen  10050
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