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Mirrors > Home > NFE Home > Th. List > eqssi | GIF version |
Description: Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.) |
Ref | Expression |
---|---|
eqssi.1 | ⊢ A ⊆ B |
eqssi.2 | ⊢ B ⊆ A |
Ref | Expression |
---|---|
eqssi | ⊢ A = B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqssi.1 | . 2 ⊢ A ⊆ B | |
2 | eqssi.2 | . 2 ⊢ B ⊆ A | |
3 | eqss 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
4 | 1, 2, 3 | mpbir2an 886 | 1 ⊢ A = B |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: inv1 3577 unv 3578 intab 3956 evenoddnnnul 4514 dmv 4920 0ima 5014 clos10 5887 cenc 6181 |
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