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Mirrors > Home > ILE 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 3112 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
4 | 1, 2, 3 | mpbir2an 926 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: inv1 3399 unv 3400 undifabs 3439 intab 3800 intid 4146 find 4513 limom 4527 dmv 4755 0ima 4899 rnxpid 4973 dftpos4 6160 axaddf 7676 axmulf 7677 dfuzi 9161 unirnioo 9756 txuni2 12425 dvef 12856 |
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