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Theorem leweon 10058

Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10059, 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 7801	. 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 6927	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
4	3	epeli 5595	. . . . . . 7 ⊢ ((1^st ‘𝑥) E (1^st ‘𝑦) ↔ (1^st ‘𝑥) ∈ (1^st ‘𝑦))
5		fvex 6927	. . . . . . . . 9 ⊢ (2^nd ‘𝑦) ∈ V
6	5	epeli 5595	. . . . . . . 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 915	. . . . . 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 5218	. . . 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 2768	. . 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 8163	. 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 848 = wceq 1539 ∈ wcel 2108 class class class wbr 5151 {copab 5213 E cep 5592 We wwe 5644 × cxp 5691 Oncon0 6392 ‘cfv 6569 1^st c1st 8020 2^nd c2nd 8021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pr 5441 ax-un 7761

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3483 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-int 4955 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-ord 6395 df-on 6396 df-iota 6522 df-fun 6571 df-fv 6577 df-1st 8022 df-2nd 8023

This theorem is referenced by: r0weon 10059

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