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Mirrors > Home > MPE Home > Th. List > wexp | Structured version Visualization version GIF version |
Description: A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
Ref | Expression |
---|---|
wexp.1 | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} |
Ref | Expression |
---|---|
wexp | ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 5678 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | wefr 5678 | . . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Fr 𝐵) | |
3 | wexp.1 | . . . 4 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} | |
4 | 3 | frxp 8149 | . . 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 5679 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
7 | weso 5679 | . . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵) | |
8 | 3 | soxp 8152 | . . 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 5642 | . 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 1536 ∈ wcel 2105 class class class wbr 5147 {copab 5209 Or wor 5595 Fr wfr 5637 We wwe 5639 × cxp 5686 ‘cfv 6562 1st c1st 8010 2nd c2nd 8011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fv 6570 df-1st 8012 df-2nd 8013 |
This theorem is referenced by: fnwelem 8154 leweon 10048 |
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