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Theorem r0weon 9427
Description: A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-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)))}
Assertion
Ref Expression
r0weon (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Distinct variable groups:   𝑧,𝑤,𝐿   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧,𝑤)   𝐿(𝑥,𝑦)

Proof of Theorem r0weon
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 r0weon.1 . . . . 5 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
2 fveq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
3 fveq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (2nd𝑥) = (2nd𝑧))
42, 3uneq12d 4139 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑧) ∪ (2nd𝑧)))
5 eqid 2821 . . . . . . . . . . 11 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
6 fvex 6677 . . . . . . . . . . . 12 (1st𝑧) ∈ V
7 fvex 6677 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
86, 7unex 7457 . . . . . . . . . . 11 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
94, 5, 8fvmpt 6762 . . . . . . . . . 10 (𝑧 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((1st𝑧) ∪ (2nd𝑧)))
10 fveq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (1st𝑥) = (1st𝑤))
11 fveq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (2nd𝑥) = (2nd𝑤))
1210, 11uneq12d 4139 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑤) ∪ (2nd𝑤)))
13 fvex 6677 . . . . . . . . . . . 12 (1st𝑤) ∈ V
14 fvex 6677 . . . . . . . . . . . 12 (2nd𝑤) ∈ V
1513, 14unex 7457 . . . . . . . . . . 11 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
1612, 5, 15fvmpt 6762 . . . . . . . . . 10 (𝑤 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) = ((1st𝑤) ∪ (2nd𝑤)))
179, 16breqan12d 5074 . . . . . . . . 9 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) E ((1st𝑤) ∪ (2nd𝑤))))
1815epeli 5462 . . . . . . . . 9 (((1st𝑧) ∪ (2nd𝑧)) E ((1st𝑤) ∪ (2nd𝑤)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)))
1917, 18syl6bb 288 . . . . . . . 8 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤))))
209, 16eqeqan12d 2838 . . . . . . . . 9 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
2120anbi1d 629 . . . . . . . 8 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤) ↔ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
2219, 21orbi12d 912 . . . . . . 7 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
2322pm5.32i 575 . . . . . 6 (((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤))) ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
2423opabbii 5125 . . . . 5 {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
251, 24eqtr4i 2847 . . . 4 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))}
26 xp1st 7712 . . . . . . . 8 (𝑥 ∈ (On × On) → (1st𝑥) ∈ On)
27 xp2nd 7713 . . . . . . . 8 (𝑥 ∈ (On × On) → (2nd𝑥) ∈ On)
28 fvex 6677 . . . . . . . . . 10 (1st𝑥) ∈ V
2928elon 6194 . . . . . . . . 9 ((1st𝑥) ∈ On ↔ Ord (1st𝑥))
30 fvex 6677 . . . . . . . . . 10 (2nd𝑥) ∈ V
3130elon 6194 . . . . . . . . 9 ((2nd𝑥) ∈ On ↔ Ord (2nd𝑥))
32 ordun 6286 . . . . . . . . 9 ((Ord (1st𝑥) ∧ Ord (2nd𝑥)) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3329, 31, 32syl2anb 597 . . . . . . . 8 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3426, 27, 33syl2anc 584 . . . . . . 7 (𝑥 ∈ (On × On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3528, 30unex 7457 . . . . . . . 8 ((1st𝑥) ∪ (2nd𝑥)) ∈ V
3635elon 6194 . . . . . . 7 (((1st𝑥) ∪ (2nd𝑥)) ∈ On ↔ Ord ((1st𝑥) ∪ (2nd𝑥)))
3734, 36sylibr 235 . . . . . 6 (𝑥 ∈ (On × On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ On)
385, 37fmpti 6869 . . . . 5 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On
3938a1i 11 . . . 4 (⊤ → (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On)
40 epweon 7485 . . . . 5 E We On
4140a1i 11 . . . 4 (⊤ → E We On)
42 leweon.1 . . . . . 6 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
4342leweon 9426 . . . . 5 𝐿 We (On × On)
4443a1i 11 . . . 4 (⊤ → 𝐿 We (On × On))
45 vex 3498 . . . . . . . 8 𝑢 ∈ V
4645dmex 7604 . . . . . . 7 dom 𝑢 ∈ V
4745rnex 7605 . . . . . . 7 ran 𝑢 ∈ V
4846, 47unex 7457 . . . . . 6 (dom 𝑢 ∪ ran 𝑢) ∈ V
49 imadmres 6085 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢)
50 inss2 4205 . . . . . . . . . 10 (𝑢 ∩ (On × On)) ⊆ (On × On)
51 ssun1 4147 . . . . . . . . . . . . . 14 dom 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
52 elinel2 4172 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ (On × On))
53 1st2nd2 7719 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (On × On) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
5452, 53syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
55 elinel1 4171 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥𝑢)
5654, 55eqeltrrd 2914 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑢 ∩ (On × On)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢)
5728, 30opeldm 5770 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (1st𝑥) ∈ dom 𝑢)
5856, 57syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ dom 𝑢)
5951, 58sseldi 3964 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
60 ssun2 4148 . . . . . . . . . . . . . 14 ran 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
6128, 30opelrn 5807 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (2nd𝑥) ∈ ran 𝑢)
6256, 61syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ ran 𝑢)
6360, 62sseldi 3964 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
6459, 63prssd 4749 . . . . . . . . . . . 12 (𝑥 ∈ (𝑢 ∩ (On × On)) → {(1st𝑥), (2nd𝑥)} ⊆ (dom 𝑢 ∪ ran 𝑢))
6552, 26syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ On)
6652, 27syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ On)
67 ordunpr 7529 . . . . . . . . . . . . 13 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6865, 66, 67syl2anc 584 . . . . . . . . . . . 12 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6964, 68sseldd 3967 . . . . . . . . . . 11 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢))
7069rgen 3148 . . . . . . . . . 10 𝑥 ∈ (𝑢 ∩ (On × On))((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)
71 ssrab 4048 . . . . . . . . . 10 ((𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)} ↔ ((𝑢 ∩ (On × On)) ⊆ (On × On) ∧ ∀𝑥 ∈ (𝑢 ∩ (On × On))((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)))
7250, 70, 71mpbir2an 707 . . . . . . . . 9 (𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
73 dmres 5869 . . . . . . . . . 10 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))))
7438fdmi 6518 . . . . . . . . . . 11 dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (On × On)
7574ineq2i 4185 . . . . . . . . . 10 (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))) = (𝑢 ∩ (On × On))
7673, 75eqtri 2844 . . . . . . . . 9 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ (On × On))
775mptpreima 6086 . . . . . . . . 9 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢)) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
7872, 76, 773sstr4i 4009 . . . . . . . 8 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢))
79 funmpt 6387 . . . . . . . . 9 Fun (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
80 resss 5872 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
81 dmss 5765 . . . . . . . . . 10 (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) → dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))))
8280, 81ax-mp 5 . . . . . . . . 9 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
83 funimass3 6817 . . . . . . . . 9 ((Fun (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ∧ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢))))
8479, 82, 83mp2an 688 . . . . . . . 8 (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢)))
8578, 84mpbir 232 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢)
8649, 85eqsstrri 4001 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (dom 𝑢 ∪ ran 𝑢)
8748, 86ssexi 5218 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
8887a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
8925, 39, 41, 44, 88fnwe 7817 . . 3 (⊤ → 𝑅 We (On × On))
90 epse 5532 . . . . 5 E Se On
9190a1i 11 . . . 4 (⊤ → E Se On)
92 vuniex 7455 . . . . . . . 8 𝑢 ∈ V
9392pwex 5273 . . . . . . 7 𝒫 𝑢 ∈ V
9493, 93xpex 7464 . . . . . 6 (𝒫 𝑢 × 𝒫 𝑢) ∈ V
955mptpreima 6086 . . . . . . . 8 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢}
96 df-rab 3147 . . . . . . . 8 {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢} = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9795, 96eqtri 2844 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9853adantr 481 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
99 elssuni 4861 . . . . . . . . . . . . 13 (((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢 → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
10099adantl 482 . . . . . . . . . . . 12 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
101100unssad 4162 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ⊆ 𝑢)
10228elpw 4544 . . . . . . . . . . 11 ((1st𝑥) ∈ 𝒫 𝑢 ↔ (1st𝑥) ⊆ 𝑢)
103101, 102sylibr 235 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ∈ 𝒫 𝑢)
104100unssbd 4163 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ⊆ 𝑢)
10530elpw 4544 . . . . . . . . . . 11 ((2nd𝑥) ∈ 𝒫 𝑢 ↔ (2nd𝑥) ⊆ 𝑢)
106104, 105sylibr 235 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ∈ 𝒫 𝑢)
107103, 106jca 512 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢))
108 elxp6 7714 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢)))
10998, 107, 108sylanbrc 583 . . . . . . . 8 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢))
110109abssi 4045 . . . . . . 7 {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)} ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11197, 110eqsstri 4000 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11294, 111ssexi 5218 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
113112a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
11425, 39, 91, 113fnse 7818 . . 3 (⊤ → 𝑅 Se (On × On))
11589, 114jca 512 . 2 (⊤ → (𝑅 We (On × On) ∧ 𝑅 Se (On × On)))
116115mptru 1535 1 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841   = wceq 1528  wtru 1529  wcel 2105  {cab 2799  wral 3138  {crab 3142  Vcvv 3495  cun 3933  cin 3934  wss 3935  𝒫 cpw 4537  {cpr 4561  cop 4565   cuni 4832   class class class wbr 5058  {copab 5120  cmpt 5138   E cep 5458   Se wse 5506   We wwe 5507   × cxp 5547  ccnv 5548  dom cdm 5549  ran crn 5550  cres 5551  cima 5552  Ord word 6184  Oncon0 6185  Fun wfun 6343  wf 6345  cfv 6349  1st c1st 7678  2nd c2nd 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-1st 7680  df-2nd 7681
This theorem is referenced by:  infxpenlem  9428
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