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Theorem wexp 8105

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 5635	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2		wefr 5635	. . 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 8101	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
5	1, 2, 4	syl2an 605	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
6		weso 5636	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
7		weso 5636	. . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵)
8	3	soxp 8104	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
9	6, 7, 8	syl2an 605	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵))
10		df-we 5600	. 2 ⊢ (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵)))
11	5, 9, 10	sylanbrc 592	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 {copab 5161 Or wor 5552 Fr wfr 5595 We wwe 5597 × cxp 5643 ‘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-nul 4286 df-if 4480 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-id 5540 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-iota 6473 df-fun 6519 df-fv 6525 df-1st 7966 df-2nd 7967

This theorem is referenced by: fnwelem 8106 leweon 9964

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