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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 9651, 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 7579 | . 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 6749 | . . . . . . . 8 ⊢ (1st ‘𝑦) ∈ V | |
4 | 3 | epeli 5477 | . . . . . . 7 ⊢ ((1st ‘𝑥) E (1st ‘𝑦) ↔ (1st ‘𝑥) ∈ (1st ‘𝑦)) |
5 | fvex 6749 | . . . . . . . . 9 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | epeli 5477 | . . . . . . . 8 ⊢ ((2nd ‘𝑥) E (2nd ‘𝑦) ↔ (2nd ‘𝑥) ∈ (2nd ‘𝑦)) |
7 | 6 | anbi2i 626 | . . . . . . 7 ⊢ (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦)) ↔ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))) |
8 | 4, 7 | orbi12i 915 | . . . . . 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 626 | . . . . 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 5135 | . . . 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 2769 | . . 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 7918 | . 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On)) |
13 | 1, 1, 12 | mp2an 692 | 1 ⊢ 𝐿 We (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2111 class class class wbr 5068 {copab 5130 E cep 5474 We wwe 5523 × cxp 5564 Oncon0 6231 ‘cfv 6398 1st c1st 7778 2nd c2nd 7779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-ord 6234 df-on 6235 df-iota 6356 df-fun 6400 df-fv 6406 df-1st 7780 df-2nd 7781 |
This theorem is referenced by: r0weon 9651 |
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