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Mirrors > Home > ILE Home > Th. List > dfss5 | GIF version |
Description: Another definition of subclasshood. Similar to df-ss 3157, dfss 3158, and dfss1 3354. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
dfss5 | ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss1 3354 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
2 | eqcom 2191 | . 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐴 ↔ 𝐴 = (𝐵 ∩ 𝐴)) | |
3 | 1, 2 | bitri 184 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∩ cin 3143 ⊆ wss 3144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 |
This theorem is referenced by: nninfdcex 11985 nnmindc 12066 nnminle 12067 |
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