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Mirrors > Home > ILE Home > Th. List > disj2 | GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
disj2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3197 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 3494 | . 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 Vcvv 2756 ∖ cdif 3146 ∩ cin 3148 ⊆ wss 3149 ∅c0 3442 |
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-in1 615 ax-in2 616 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-ral 2473 df-v 2758 df-dif 3151 df-in 3155 df-ss 3162 df-nul 3443 |
This theorem is referenced by: ssindif0im 3502 intirr 5040 setsresg 12630 setscom 12632 |
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