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Mirrors > Home > MPE Home > Th. List > leweon | Structured version Visualization version GIF version |
Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10036, this order is not set-like, as the preimage of ⟨1o, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.) |
Ref | Expression |
---|---|
leweon.1 | ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} |
Ref | Expression |
---|---|
leweon | ⊢ 𝐿 We (On × On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epweon 7777 | . 2 ⊢ E We On | |
2 | leweon.1 | . . . 4 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} | |
3 | fvex 6910 | . . . . . . . 8 ⊢ (1st ‘𝑦) ∈ V | |
4 | 3 | epeli 5584 | . . . . . . 7 ⊢ ((1st ‘𝑥) E (1st ‘𝑦) ↔ (1st ‘𝑥) ∈ (1st ‘𝑦)) |
5 | fvex 6910 | . . . . . . . . 9 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | epeli 5584 | . . . . . . . 8 ⊢ ((2nd ‘𝑥) E (2nd ‘𝑦) ↔ (2nd ‘𝑥) ∈ (2nd ‘𝑦)) |
7 | 6 | anbi2i 622 | . . . . . . 7 ⊢ (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦)) ↔ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))) |
8 | 4, 7 | orbi12i 913 | . . . . . 6 ⊢ (((1st ‘𝑥) E (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦))) ↔ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦)))) |
9 | 8 | anbi2i 622 | . . . . 5 ⊢ (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) E (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦)))) ↔ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))) |
10 | 9 | opabbii 5215 | . . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) E (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} |
11 | 2, 10 | eqtr4i 2759 | . . 3 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) E (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦))))} |
12 | 11 | wexp 8135 | . 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On)) |
13 | 1, 1, 12 | mp2an 691 | 1 ⊢ 𝐿 We (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 {copab 5210 E cep 5581 We wwe 5632 × cxp 5676 Oncon0 6369 ‘cfv 6548 1st c1st 7991 2nd c2nd 7992 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 df-2nd 7994 |
This theorem is referenced by: r0weon 10036 |
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