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Theorem leweon 9909

Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9910, this order is not set-like, as the preimage of ⟨1_o, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.)

**Hypothesis**
Ref	Expression
leweon.1	⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}

**Assertion**
Ref	Expression
leweon	⊢ 𝐿 We (On × On)

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝐿(𝑥,𝑦)

**Proof of Theorem leweon**
Step	Hyp	Ref	Expression
1		epweon 7714	. 2 ⊢ E We On
2		leweon.1	. . . 4 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
3		fvex 6841	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
4	3	epeli 5521	. . . . . . 7 ⊢ ((1^st ‘𝑥) E (1^st ‘𝑦) ↔ (1^st ‘𝑥) ∈ (1^st ‘𝑦))
5		fvex 6841	. . . . . . . . 9 ⊢ (2^nd ‘𝑦) ∈ V
6	5	epeli 5521	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ↔ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))
7	6	anbi2i 623	. . . . . . 7 ⊢ (((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦)) ↔ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦)))
8	4, 7	orbi12i 914	. . . . . 6 ⊢ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))) ↔ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))
9	8	anbi2i 623	. . . . 5 ⊢ (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦)))) ↔ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦)))))
10	9	opabbii 5160	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
11	2, 10	eqtr4i 2759	. . 3 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))))}
12	11	wexp 8066	. 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 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 class class class wbr 5093 {copab 5155 E cep 5518 We wwe 5571 × cxp 5617 Oncon0 6311 ‘cfv 6486 1^st c1st 7925 2^nd c2nd 7926

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6314 df-on 6315 df-iota 6442 df-fun 6488 df-fv 6494 df-1st 7927 df-2nd 7928

This theorem is referenced by: r0weon 9910

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