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Mirrors > Home > MPE Home > Th. List > disj1 | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.) |
Ref | Expression |
---|---|
disj1 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 4401 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
2 | df-ral 3145 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 277 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-dif 3941 df-in 3945 df-nul 4294 |
This theorem is referenced by: reldisj 4404 disj3 4405 undif4 4418 disjsn 4649 funun 6402 zfregs2 9177 dfac5lem4 9554 isf32lem9 9785 fzodisj 13074 fzodisjsn 13078 inpr0 30294 bnj1280 32294 ecin0 35608 zfregs2VD 41182 |
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