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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 5293 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
2 | soss 5293 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
3 | 1, 2 | anim12i 606 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) |
4 | eqss 3836 | . . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | dfbi2 468 | . . 3 ⊢ ((𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) | |
6 | 3, 4, 5 | 3imtr4i 284 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴)) |
7 | 6 | bicomd 215 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ⊆ wss 3792 Or wor 5273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-ral 3095 df-in 3799 df-ss 3806 df-po 5274 df-so 5275 |
This theorem is referenced by: weeq2 5344 wemapso2 8747 oemapso 8876 fin2i 9452 isfin2-2 9476 fin1a2lem10 9566 zorn2lem7 9659 zornn0g 9662 opsrtoslem2 19881 sltsolem1 32415 soeq12d 38571 aomclem1 38587 |
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