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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 9438, 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 7497 | . 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 6683 | . . . . . . . 8 ⊢ (1st ‘𝑦) ∈ V | |
4 | 3 | epeli 5468 | . . . . . . 7 ⊢ ((1st ‘𝑥) E (1st ‘𝑦) ↔ (1st ‘𝑥) ∈ (1st ‘𝑦)) |
5 | fvex 6683 | . . . . . . . . 9 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | epeli 5468 | . . . . . . . 8 ⊢ ((2nd ‘𝑥) E (2nd ‘𝑦) ↔ (2nd ‘𝑥) ∈ (2nd ‘𝑦)) |
7 | 6 | anbi2i 624 | . . . . . . 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 624 | . . . . 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 5133 | . . . 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 2847 | . . 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 7824 | . 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On)) |
13 | 1, 1, 12 | mp2an 690 | 1 ⊢ 𝐿 We (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 {copab 5128 E cep 5464 We wwe 5513 × cxp 5553 Oncon0 6191 ‘cfv 6355 1st c1st 7687 2nd c2nd 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: r0weon 9438 |
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