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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 5675 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
| 2 | wefr 5675 | . . 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 8151 | . . 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 5676 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
| 7 | weso 5676 | . . 3 ⊢ (𝑆 We 𝐵 → 𝑆 Or 𝐵) | |
| 8 | 3 | soxp 8154 | . . 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 5639 | . 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 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 {copab 5205 Or wor 5591 Fr wfr 5634 We wwe 5636 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: fnwelem 8156 leweon 10051 |
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