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Theorem leweon 9964

Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9965, 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 7754	. 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 6876	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
4	3	epeli 5547	. . . . . . 7 ⊢ ((1^st ‘𝑥) E (1^st ‘𝑦) ↔ (1^st ‘𝑥) ∈ (1^st ‘𝑦))
5		fvex 6876	. . . . . . . . 9 ⊢ (2^nd ‘𝑦) ∈ V
6	5	epeli 5547	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ↔ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))
7	6	anbi2i 632	. . . . . . 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 925	. . . . . 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 632	. . . . 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 5166	. . . 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 2787	. . 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 8105	. 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On))
13	1, 1, 12	mp2an 702	1 ⊢ 𝐿 We (On × On)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 {copab 5161 E cep 5544 We wwe 5597 × cxp 5643 Oncon0 6342 ‘cfv 6517 1^st c1st 7964 2^nd c2nd 7965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-ord 6345 df-on 6346 df-iota 6473 df-fun 6519 df-fv 6525 df-1st 7966 df-2nd 7967

This theorem is referenced by: r0weon 9965

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