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Theorem wexp 8112

Description: A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

**Hypothesis**
Ref	Expression
wexp.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥)𝑆(2^nd ‘𝑦))))}

**Assertion**
Ref	Expression
wexp	⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝑅,𝑦 𝑥,𝑆,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦)

**Proof of Theorem wexp**
Step	Hyp	Ref	Expression
1		wefr 5631	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2		wefr 5631	. . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Fr 𝐵)
3		wexp.1	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥)𝑆(2^nd ‘𝑦))))}
4	3	frxp 8108	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
5	1, 2, 4	syl2an 596	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
6		weso 5632	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
7		weso 5632	. . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵)
8	3	soxp 8111	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
9	6, 7, 8	syl2an 596	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵))
10		df-we 5596	. 2 ⊢ (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵)))
11	5, 9, 10	sylanbrc 583	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 class class class wbr 5110 {copab 5172 Or wor 5548 Fr wfr 5591 We wwe 5593 × cxp 5639 ‘cfv 6514 1^st c1st 7969 2^nd c2nd 7970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fv 6522 df-1st 7971 df-2nd 7972

This theorem is referenced by: fnwelem 8113 leweon 9971

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