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Mirrors > Home > MPE Home > Th. List > soeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
Ref | Expression |
---|---|
soeq2 | ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soss 4977 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
2 | soss 4977 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
3 | 1, 2 | anim12i 588 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) |
4 | eqss 3583 | . . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | dfbi2 658 | . . 3 ⊢ ((𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) | |
6 | 3, 4, 5 | 3imtr4i 280 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴)) |
7 | 6 | bicomd 212 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊆ wss 3540 Or wor 4958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ral 2901 df-in 3547 df-ss 3554 df-po 4959 df-so 4960 |
This theorem is referenced by: weeq2 5027 wemapso2 8341 oemapso 8462 fin2i 9000 isfin2-2 9024 fin1a2lem10 9114 zorn2lem7 9207 zornn0g 9210 opsrtoslem2 19306 sltsolem1 31067 soeq12d 36626 aomclem1 36642 |
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