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Theorem leweon 9045

Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9046, 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 7150	. 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 6364	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
4	3	epelc 5182	. . . . . . 7 ⊢ ((1^st ‘𝑥) E (1^st ‘𝑦) ↔ (1^st ‘𝑥) ∈ (1^st ‘𝑦))
5		fvex 6364	. . . . . . . . 9 ⊢ (2^nd ‘𝑦) ∈ V
6	5	epelc 5182	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ↔ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))
7	6	anbi2i 732	. . . . . . 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 544	. . . . . 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 732	. . . . 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 4870	. . . 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 2786	. . 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 7461	. 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On))
13	1, 1, 12	mp2an 710	1 ⊢ 𝐿 We (On × On)

Colors of variables: wff setvar class

Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2140 class class class wbr 4805 {copab 4865 E cep 5179 We wwe 5225 × cxp 5265 Oncon0 5885 ‘cfv 6050 1^st c1st 7333 2^nd c2nd 7334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-ord 5888 df-on 5889 df-iota 6013 df-fun 6052 df-fv 6058 df-1st 7335 df-2nd 7336

This theorem is referenced by: r0weon 9046

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