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Theorem infxpenlem 10053
Description: Lemma for infxpen 10054. (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 4010 . . . 4 (𝑎 = 𝑚 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝑚))
2 xpeq12 5710 . . . . . 6 ((𝑎 = 𝑚𝑎 = 𝑚) → (𝑎 × 𝑎) = (𝑚 × 𝑚))
32anidms 566 . . . . 5 (𝑎 = 𝑚 → (𝑎 × 𝑎) = (𝑚 × 𝑚))
4 id 22 . . . . 5 (𝑎 = 𝑚𝑎 = 𝑚)
53, 4breq12d 5156 . . . 4 (𝑎 = 𝑚 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝑚 × 𝑚) ≈ 𝑚))
61, 5imbi12d 344 . . 3 (𝑎 = 𝑚 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)))
7 sseq2 4010 . . . 4 (𝑎 = 𝐴 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝐴))
8 xpeq12 5710 . . . . . 6 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑎 × 𝑎) = (𝐴 × 𝐴))
98anidms 566 . . . . 5 (𝑎 = 𝐴 → (𝑎 × 𝑎) = (𝐴 × 𝐴))
10 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
119, 10breq12d 5156 . . . 4 (𝑎 = 𝐴 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝐴 × 𝐴) ≈ 𝐴))
127, 11imbi12d 344 . . 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 3484 . . . . . . . . . . . . 13 𝑎 ∈ V
1514, 14xpex 7773 . . . . . . . . . . . 12 (𝑎 × 𝑎) ∈ V
16 simpll 767 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
1713, 16sylbi 217 . . . . . . . . . . . . . . . . 17 (𝜑𝑎 ∈ On)
18 onss 7805 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ On → 𝑎 ⊆ On)
1917, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑎 ⊆ On)
20 xpss12 5700 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ On ∧ 𝑎 ⊆ On) → (𝑎 × 𝑎) ⊆ (On × On))
2119, 19, 20syl2anc 584 . . . . . . . . . . . . . . 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 10052 . . . . . . . . . . . . . . . 16 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
2524simpli 483 . . . . . . . . . . . . . . 15 𝑅 We (On × On)
26 wess 5671 . . . . . . . . . . . . . . 15 ((𝑎 × 𝑎) ⊆ (On × On) → (𝑅 We (On × On) → 𝑅 We (𝑎 × 𝑎)))
2721, 25, 26mpisyl 21 . . . . . . . . . . . . . 14 (𝜑𝑅 We (𝑎 × 𝑎))
28 weinxp 5770 . . . . . . . . . . . . . 14 (𝑅 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
2927, 28sylib 218 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
30 infxpen.1 . . . . . . . . . . . . . 14 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
31 weeq1 5672 . . . . . . . . . . . . . 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 234 . . . . . . . . . . . 12 (𝜑𝑄 We (𝑎 × 𝑎))
34 infxpen.4 . . . . . . . . . . . . 13 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
3534oiiso 9577 . . . . . . . . . . . 12 (((𝑎 × 𝑎) ∈ V ∧ 𝑄 We (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
3615, 33, 35sylancr 587 . . . . . . . . . . 11 (𝜑𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
37 isof1o 7343 . . . . . . . . . . 11 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎))
38 f1ocnv 6860 . . . . . . . . . . 11 (𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎) → 𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽)
39 f1of1 6847 . . . . . . . . . . 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 6859 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽)
4215f1oen 9013 . . . . . . . . . 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 6849 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽 Fn (𝑎 × 𝑎))
4536, 37, 38, 444syl 19 . . . . . . . . . 10 (𝜑𝐽 Fn (𝑎 × 𝑎))
4636adantr 480 . . . . . . . . . . . . . . . . 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 6100 . . . . . . . . . . . . . . . . . . 19 (𝑄 “ {𝑤}) ⊆ dom 𝑄
49 inss2 4238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
5030, 49eqsstri 4030 . . . . . . . . . . . . . . . . . . . . 21 𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
51 dmss 5913 . . . . . . . . . . . . . . . . . . . . 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 5941 . . . . . . . . . . . . . . . . . . . 20 dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)) = (𝑎 × 𝑎)
5452, 53sseqtri 4032 . . . . . . . . . . . . . . . . . . 19 dom 𝑄 ⊆ (𝑎 × 𝑎)
5548, 54sstri 3993 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)
56 f1ores 6862 . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . 17 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5815, 15xpex 7773 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 × 𝑎) × (𝑎 × 𝑎)) ∈ V
5958inex2 5318 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ∈ V
6030, 59eqeltri 2837 . . . . . . . . . . . . . . . . . . . 20 𝑄 ∈ V
6160cnvex 7947 . . . . . . . . . . . . . . . . . . 19 𝑄 ∈ V
6261imaex 7936 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ∈ V
6362f1oen 9013 . . . . . . . . . . . . . . . . 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 4223 . . . . . . . . . . . . . . . . . . 19 ((𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎) ↔ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤}))
6655, 65mpbi 230 . . . . . . . . . . . . . . . . . 18 ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤})
6766imaeq2i 6076 . . . . . . . . . . . . . . . . 17 (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽 “ (𝑄 “ {𝑤}))
68 isocnv 7350 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑤 ∈ (𝑎 × 𝑎))
71 isoini 7358 . . . . . . . . . . . . . . . . . . 19 ((𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽) ∧ 𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
7269, 70, 71syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
73 fvex 6919 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝑤) ∈ V
7473epini 6114 . . . . . . . . . . . . . . . . . . . 20 ( E “ {(𝐽𝑤)}) = (𝐽𝑤)
7574ineq2i 4217 . . . . . . . . . . . . . . . . . . 19 (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (dom 𝐽 ∩ (𝐽𝑤))
7634oicl 9569 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐽
77 f1of 6848 . . . . . . . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ dom 𝐽)
80 ordelss 6400 . . . . . . . . . . . . . . . . . . . . 21 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ⊆ dom 𝐽)
8176, 79, 80sylancr 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ⊆ dom 𝐽)
82 sseqin2 4223 . . . . . . . . . . . . . . . . . . . 20 ((𝐽𝑤) ⊆ dom 𝐽 ↔ (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8381, 82sylib 218 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8475, 83eqtrid 2789 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (𝐽𝑤))
8572, 84eqtrd 2777 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽𝑤))
8667, 85eqtr3id 2791 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ (𝑄 “ {𝑤})) = (𝐽𝑤))
8764, 86breqtrd 5169 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽𝑤))
8887ensymd 9045 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≈ (𝑄 “ {𝑤}))
89 infxpen.3 . . . . . . . . . . . . . . . . . . 19 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
90 fvex 6919 . . . . . . . . . . . . . . . . . . . 20 (1st𝑤) ∈ V
91 fvex 6919 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑤) ∈ V
9290, 91unex 7764 . . . . . . . . . . . . . . . . . . 19 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
9389, 92eqeltri 2837 . . . . . . . . . . . . . . . . . 18 𝑀 ∈ V
9493sucex 7826 . . . . . . . . . . . . . . . . 17 suc 𝑀 ∈ V
9594, 94xpex 7773 . . . . . . . . . . . . . . . 16 (suc 𝑀 × suc 𝑀) ∈ V
96 xpss 5701 . . . . . . . . . . . . . . . . . . . 20 (𝑎 × 𝑎) ⊆ (V × V)
97 simp3 1139 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑄 “ {𝑤}))
98 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ V
9998eliniseg 6112 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ V → (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤))
10099elv 3485 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤)
10197, 100sylib 218 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧𝑄𝑤)
10230breqi 5149 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑄𝑤𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤)
103 brin 5195 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
104102, 103bitri 275 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑄𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
105104simprbi 496 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤)
106 brxp 5734 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤 ↔ (𝑧 ∈ (𝑎 × 𝑎) ∧ 𝑤 ∈ (𝑎 × 𝑎)))
107106simplbi 497 . . . . . . . . . . . . . . . . . . . . 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 3981 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (V × V))
11017adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ On)
1111103adant3 1133 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑎 ∈ On)
112 xp1st 8046 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (1st𝑧) ∈ 𝑎)
113 onelon 6409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (1st𝑧) ∈ 𝑎) → (1st𝑧) ∈ On)
114112, 113sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (1st𝑧) ∈ On)
115111, 108, 114syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ On)
116 eloni 6394 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ On → Ord 𝑎)
117 elxp7 8049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ (𝑎 × 𝑎) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎)))
118117simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎))
119 ordunel 7847 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝑎 ∧ (1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
1201193expib 1123 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2845 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑀𝑎)
124 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 𝑚𝑎)
12513, 124sylbi 217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∀𝑚𝑎 𝑚𝑎)
126 simprl 771 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
12713, 126sylbi 217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ω ⊆ 𝑎)
128 iscard 10015 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 ↔ (𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎))
129 cardlim 10012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎))
130 sseq2 4010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (ω ⊆ (card‘𝑎) ↔ ω ⊆ 𝑎))
131 limeq 6396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (Lim (card‘𝑎) ↔ Lim 𝑎))
132130, 131bibi12d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((card‘𝑎) = 𝑎 → ((ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎)) ↔ (ω ⊆ 𝑎 ↔ Lim 𝑎)))
133129, 132mpbii 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 → (ω ⊆ 𝑎 ↔ Lim 𝑎))
134128, 133sylbir 235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → (ω ⊆ 𝑎 ↔ Lim 𝑎))
135134biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) ∧ ω ⊆ 𝑎) → Lim 𝑎)
13617, 125, 127, 135syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → Lim 𝑎)
137136adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → Lim 𝑎)
138 limsuc 7870 . . . . . . . . . . . . . . . . . . . . . . . 24 (Lim 𝑎 → (𝑀𝑎 ↔ suc 𝑀𝑎))
139137, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑀𝑎 ↔ suc 𝑀𝑎))
140123, 139mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
141 onelon 6409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ suc 𝑀𝑎) → suc 𝑀 ∈ On)
14217, 140, 141syl2an2r 685 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀 ∈ On)
1431423adant3 1133 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → suc 𝑀 ∈ On)
144 ssun1 4178 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
145144a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
146104simplbi 497 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧𝑅𝑤)
147 df-br 5144 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧𝑅𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑅)
14823eleq2i 2833 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5529 . . . . . . . . . . . . . . . . . . . . . . . . . 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 297 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧𝑅𝑤 ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
151150simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
152 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤) → ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)))
153152orim2i 911 . . . . . . . . . . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1st𝑧) ∈ V
156 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2nd𝑧) ∈ V
157155, 156unex 7764 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
158157elsuc 6454 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
159154, 158sylibr 234 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)))
160 suceq 6450 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 = ((1st𝑤) ∪ (2nd𝑤)) → suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤)))
16189, 160ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤))
162159, 161eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
163101, 146, 1623syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
164 ontr2 6431 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (1st𝑧) ∈ suc 𝑀))
165164imp 406 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (1st𝑧) ∈ suc 𝑀)
166115, 143, 145, 163, 165syl22anc 839 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ suc 𝑀)
167 xp2nd 8047 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (2nd𝑧) ∈ 𝑎)
168 onelon 6409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (2nd𝑧) ∈ 𝑎) → (2nd𝑧) ∈ On)
169167, 168sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (2nd𝑧) ∈ On)
170111, 108, 169syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ On)
171 ssun2 4179 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
172171a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
173 ontr2 6431 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (2nd𝑧) ∈ suc 𝑀))
174173imp 406 . . . . . . . . . . . . . . . . . . . 20 ((((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (2nd𝑧) ∈ suc 𝑀)
175170, 143, 172, 163, 174syl22anc 839 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ suc 𝑀)
176 elxp7 8049 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (suc 𝑀 × suc 𝑀) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)))
177176biimpri 228 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
178109, 166, 175, 177syl12anc 837 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
1791783expia 1122 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑧 ∈ (𝑄 “ {𝑤}) → 𝑧 ∈ (suc 𝑀 × suc 𝑀)))
180179ssrdv 3989 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀))
181 ssdomg 9040 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ω ⊆ 𝑎)
184 nnfi 9207 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
185 xpfi 9358 . . . . . . . . . . . . . . . . . . . . . 22 ((suc 𝑀 ∈ Fin ∧ suc 𝑀 ∈ Fin) → (suc 𝑀 × suc 𝑀) ∈ Fin)
186185anidms 566 . . . . . . . . . . . . . . . . . . . . 21 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ∈ Fin)
187 isfinite 9692 . . . . . . . . . . . . . . . . . . . . 21 ((suc 𝑀 × suc 𝑀) ∈ Fin ↔ (suc 𝑀 × suc 𝑀) ≺ ω)
188186, 187sylib 218 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ≺ ω)
189184, 188syl 17 . . . . . . . . . . . . . . . . . . 19 (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ ω)
190 ssdomg 9040 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (ω ⊆ 𝑎 → ω ≼ 𝑎))
191190elv 3485 . . . . . . . . . . . . . . . . . . 19 (ω ⊆ 𝑎 → ω ≼ 𝑎)
192 sdomdomtr 9150 . . . . . . . . . . . . . . . . . . 19 (((suc 𝑀 × suc 𝑀) ≺ ω ∧ ω ≼ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
193189, 191, 192syl2an 596 . . . . . . . . . . . . . . . . . 18 ((suc 𝑀 ∈ ω ∧ ω ⊆ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
194193expcom 413 . . . . . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . . . . 18 (𝑚 = suc 𝑀 → (𝑚𝑎 ↔ suc 𝑀𝑎))
197125adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 𝑚𝑎)
198196, 197, 140rspcdva 3623 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
199 omelon 9686 . . . . . . . . . . . . . . . . . . 19 ω ∈ On
200 ontri1 6418 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ suc 𝑀 ∈ On) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
201199, 142, 200sylancr 587 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
202 sseq2 4010 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → (ω ⊆ 𝑚 ↔ ω ⊆ suc 𝑀))
203 xpeq12 5710 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = suc 𝑀𝑚 = suc 𝑀) → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
204203anidms 566 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀 → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
205 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀𝑚 = suc 𝑀)
206204, 205breq12d 5156 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → ((𝑚 × 𝑚) ≈ 𝑚 ↔ (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
207202, 206imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑚 = suc 𝑀 → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ↔ (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀)))
208 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
20913, 208sylbi 217 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
210209adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
211207, 210, 140rspcdva 3623 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
212201, 211sylbird 260 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
213 ensdomtr 9153 . . . . . . . . . . . . . . . . . 18 (((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 ∧ suc 𝑀𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
214213expcom 413 . . . . . . . . . . . . . . . . 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 9152 . . . . . . . . . . . . . . 15 (((𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀) ∧ (suc 𝑀 × suc 𝑀) ≺ 𝑎) → (𝑄 “ {𝑤}) ≺ 𝑎)
218182, 216, 217syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≺ 𝑎)
219 ensdomtr 9153 . . . . . . . . . . . . . 14 (((𝐽𝑤) ≈ (𝑄 “ {𝑤}) ∧ (𝑄 “ {𝑤}) ≺ 𝑎) → (𝐽𝑤) ≺ 𝑎)
22088, 218, 219syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≺ 𝑎)
221 ordelon 6408 . . . . . . . . . . . . . . 15 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ∈ On)
22276, 79, 221sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ On)
223 onenon 9989 . . . . . . . . . . . . . . 15 (𝑎 ∈ On → 𝑎 ∈ dom card)
224110, 223syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ dom card)
225 cardsdomel 10014 . . . . . . . . . . . . . 14 (((𝐽𝑤) ∈ On ∧ 𝑎 ∈ dom card) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
226222, 224, 225syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
227220, 226mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ (card‘𝑎))
228 eleq2 2830 . . . . . . . . . . . . . 14 ((card‘𝑎) = 𝑎 → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
229128, 228sylbir 235 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
23017, 197, 229syl2an2r 685 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
231227, 230mpbid 232 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ 𝑎)
232231ralrimiva 3146 . . . . . . . . . 10 (𝜑 → ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎)
233 fnfvrnss 7141 . . . . . . . . . . 11 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
234 ssdomg 9040 . . . . . . . . . . 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 584 . . . . . . . . 9 (𝜑 → ran 𝐽𝑎)
237 endomtr 9052 . . . . . . . . 9 (((𝑎 × 𝑎) ≈ ran 𝐽 ∧ ran 𝐽𝑎) → (𝑎 × 𝑎) ≼ 𝑎)
23843, 236, 237syl2anc 584 . . . . . . . 8 (𝜑 → (𝑎 × 𝑎) ≼ 𝑎)
23913, 238sylbir 235 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≼ 𝑎)
240 df1o2 8513 . . . . . . . . . . . 12 1o = {∅}
241 1onn 8678 . . . . . . . . . . . 12 1o ∈ ω
242240, 241eqeltrri 2838 . . . . . . . . . . 11 {∅} ∈ ω
243 nnsdom 9694 . . . . . . . . . . 11 ({∅} ∈ ω → {∅} ≺ ω)
244 sdomdom 9020 . . . . . . . . . . 11 ({∅} ≺ ω → {∅} ≼ ω)
245242, 243, 244mp2b 10 . . . . . . . . . 10 {∅} ≼ ω
246 domtr 9047 . . . . . . . . . 10 (({∅} ≼ ω ∧ ω ≼ 𝑎) → {∅} ≼ 𝑎)
247245, 191, 246sylancr 587 . . . . . . . . 9 (ω ⊆ 𝑎 → {∅} ≼ 𝑎)
248 0ex 5307 . . . . . . . . . . . 12 ∅ ∈ V
24914, 248xpsnen 9095 . . . . . . . . . . 11 (𝑎 × {∅}) ≈ 𝑎
250249ensymi 9044 . . . . . . . . . 10 𝑎 ≈ (𝑎 × {∅})
25114xpdom2 9107 . . . . . . . . . 10 ({∅} ≼ 𝑎 → (𝑎 × {∅}) ≼ (𝑎 × 𝑎))
252 endomtr 9052 . . . . . . . . . 10 ((𝑎 ≈ (𝑎 × {∅}) ∧ (𝑎 × {∅}) ≼ (𝑎 × 𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
253250, 251, 252sylancr 587 . . . . . . . . 9 ({∅} ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎))
254247, 253syl 17 . . . . . . . 8 (ω ⊆ 𝑎𝑎 ≼ (𝑎 × 𝑎))
255254ad2antrl 728 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
256 sbth 9133 . . . . . . 7 (((𝑎 × 𝑎) ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
257239, 255, 256syl2anc 584 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
258257expr 456 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
259 simplr 769 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
260 simpll 767 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
261 simprr 773 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ¬ ∀𝑚𝑎 𝑚𝑎)
262 rexnal 3100 . . . . . . . . . 10 (∃𝑚𝑎 ¬ 𝑚𝑎 ↔ ¬ ∀𝑚𝑎 𝑚𝑎)
263 onelss 6426 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
264 ssdomg 9040 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
265263, 264syld 47 . . . . . . . . . . . 12 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
266 bren2 9023 . . . . . . . . . . . . 13 (𝑚𝑎 ↔ (𝑚𝑎 ∧ ¬ 𝑚𝑎))
267266simplbi2 500 . . . . . . . . . . . 12 (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎))
268265, 267syl6 35 . . . . . . . . . . 11 (𝑎 ∈ On → (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎)))
269268reximdvai 3165 . . . . . . . . . 10 (𝑎 ∈ On → (∃𝑚𝑎 ¬ 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
270262, 269biimtrrid 243 . . . . . . . . 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 3114 . . . . . . . 8 ((∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ ∃𝑚𝑎 𝑚𝑎) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
273259, 271, 272syl2anc 584 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
274 simprl 771 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
275 onelon 6409 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ 𝑚𝑎) → 𝑚 ∈ On)
276 ensym 9043 . . . . . . . . . . . . . . . . . 18 (𝑚𝑎𝑎𝑚)
277 domentr 9053 . . . . . . . . . . . . . . . . . 18 ((ω ≼ 𝑎𝑎𝑚) → ω ≼ 𝑚)
278191, 276, 277syl2an 596 . . . . . . . . . . . . . . . . 17 ((ω ⊆ 𝑎𝑚𝑎) → ω ≼ 𝑚)
279 domnsym 9139 . . . . . . . . . . . . . . . . . . 19 (ω ≼ 𝑚 → ¬ 𝑚 ≺ ω)
280 nnsdom 9694 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ω → 𝑚 ≺ ω)
281279, 280nsyl 140 . . . . . . . . . . . . . . . . . 18 (ω ≼ 𝑚 → ¬ 𝑚 ∈ ω)
282 ontri1 6418 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ 𝑚 ∈ On) → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
283199, 282mpan 690 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
284281, 283imbitrrid 246 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ On → (ω ≼ 𝑚 → ω ⊆ 𝑚))
285275, 278, 284syl2im 40 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑚𝑎) → ((ω ⊆ 𝑎𝑚𝑎) → ω ⊆ 𝑚))
286285expd 415 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑚𝑎) → (ω ⊆ 𝑎 → (𝑚𝑎 → ω ⊆ 𝑚)))
287286impcom 407 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑚 × 𝑚) ≈ 𝑚)
290 entr 9046 . . . . . . . . . . . . . . . 16 (((𝑚 × 𝑚) ≈ 𝑚𝑚𝑎) → (𝑚 × 𝑚) ≈ 𝑎)
291290ancoms 458 . . . . . . . . . . . . . . 15 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑚 × 𝑚) ≈ 𝑎)
292 xpen 9180 . . . . . . . . . . . . . . . . 17 ((𝑎𝑚𝑎𝑚) → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
293292anidms 566 . . . . . . . . . . . . . . . 16 (𝑎𝑚 → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
294 entr 9046 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑎) ≈ (𝑚 × 𝑚) ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
295293, 294sylan 580 . . . . . . . . . . . . . . 15 ((𝑎𝑚 ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
296276, 291, 295syl2an2r 685 . . . . . . . . . . . . . 14 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑎 × 𝑎) ≈ 𝑎)
297296ex 412 . . . . . . . . . . . . 13 (𝑚𝑎 → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
298297ad2antll 729 . . . . . . . . . . . 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 412 . . . . . . . . . 10 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
301300expr 456 . . . . . . . . 9 ((ω ⊆ 𝑎𝑎 ∈ On) → (𝑚𝑎 → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎)))
302301rexlimdv 3153 . . . . . . . 8 ((ω ⊆ 𝑎𝑎 ∈ On) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
303274, 260, 302syl2anc 584 . . . . . . 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 456 . . . . 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 419 . . 3 (𝑎 ∈ On → (∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎)))
3086, 12, 307tfis3 7879 . 2 (𝐴 ∈ On → (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴))
309308imp 406 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632   class class class wbr 5143  {copab 5205   E cep 5583   Se wse 5635   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562  ωcom 7887  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  OrdIsocoi 9549  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979
This theorem is referenced by:  infxpen  10054
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