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Mirrors > Home > ILE Home > Th. List > dfss1 | GIF version |
Description: A frequently-used variant of subclass definition df-ss 3124. (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
dfss1 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3124 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | incom 3309 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | 2 | eqeq1i 2172 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
4 | 1, 3 | bitri 183 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∩ cin 3110 ⊆ wss 3111 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 |
This theorem is referenced by: dfss5 3322 sseqin2 3336 onintexmid 4544 xpimasn 5046 fndmdif 5584 infiexmid 6834 ssfidc 6891 isumss 11318 znnen 12274 |
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