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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 5599 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | |
2 | soss 5599 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
3 | 1, 2 | anim12i 612 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) |
4 | eqss 3990 | . . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | dfbi2 474 | . . 3 ⊢ ((𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵 → 𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴 → 𝑅 Or 𝐵))) | |
6 | 3, 4, 5 | 3imtr4i 292 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴)) |
7 | 6 | bicomd 222 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊆ wss 3941 Or wor 5578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-in 3948 df-ss 3958 df-po 5579 df-so 5580 |
This theorem is referenced by: weeq2 5656 wemapso2 9545 oemapso 9674 fin2i 10287 isfin2-2 10311 fin1a2lem10 10401 zorn2lem7 10494 zornn0g 10497 opsrtoslem2 21948 sltsolem1 27548 soeq12d 42330 aomclem1 42346 |
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