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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 5544 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | wefr 5544 | . . 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 7819 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵)) |
5 | 1, 2, 4 | syl2an 597 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵)) |
6 | weso 5545 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
7 | weso 5545 | . . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵) | |
8 | 3 | soxp 7822 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵)) |
9 | 6, 7, 8 | syl2an 597 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵)) |
10 | df-we 5515 | . 2 ⊢ (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵))) | |
11 | 5, 9, 10 | sylanbrc 585 | 1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 {copab 5127 Or wor 5472 Fr wfr 5510 We wwe 5512 × cxp 5552 ‘cfv 6354 1st c1st 7686 2nd c2nd 7687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-1st 7688 df-2nd 7689 |
This theorem is referenced by: fnwelem 7824 leweon 9436 |
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