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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 8718, 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 6875 | . 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 6113 | . . . . . . . 8 ⊢ (1st ‘𝑦) ∈ V | |
4 | 3 | epelc 4951 | . . . . . . 7 ⊢ ((1st ‘𝑥) E (1st ‘𝑦) ↔ (1st ‘𝑥) ∈ (1st ‘𝑦)) |
5 | fvex 6113 | . . . . . . . . 9 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | epelc 4951 | . . . . . . . 8 ⊢ ((2nd ‘𝑥) E (2nd ‘𝑦) ↔ (2nd ‘𝑥) ∈ (2nd ‘𝑦)) |
7 | 6 | anbi2i 726 | . . . . . . 7 ⊢ (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) E (2nd ‘𝑦)) ↔ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))) |
8 | 4, 7 | orbi12i 542 | . . . . . 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 726 | . . . . 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 4649 | . . . 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 2635 | . . 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 7178 | . 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On)) |
13 | 1, 1, 12 | mp2an 704 | 1 ⊢ 𝐿 We (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 {copab 4642 E cep 4947 We wwe 4996 × cxp 5036 Oncon0 5640 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: r0weon 8718 |
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