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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 10006, 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 7758 | . 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 6897 | . . . . . . . 8 ⊢ (1st ‘𝑦) ∈ V | |
4 | 3 | epeli 5575 | . . . . . . 7 ⊢ ((1st ‘𝑥) E (1st ‘𝑦) ↔ (1st ‘𝑥) ∈ (1st ‘𝑦)) |
5 | fvex 6897 | . . . . . . . . 9 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | epeli 5575 | . . . . . . . 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 911 | . . . . . 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 5208 | . . . 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 2757 | . . 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 8113 | . 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On)) |
13 | 1, 1, 12 | mp2an 689 | 1 ⊢ 𝐿 We (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 {copab 5203 E cep 5572 We wwe 5623 × cxp 5667 Oncon0 6357 ‘cfv 6536 1st c1st 7969 2nd c2nd 7970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: r0weon 10006 |
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