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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 4447 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
2 | df-ral 3061 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 275 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∩ cin 3947 ∅c0 4322 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-dif 3951 df-in 3955 df-nul 4323 |
This theorem is referenced by: reldisj 4451 reldisjOLD 4452 disj3 4453 undif4 4466 disjsn 4715 funun 6594 zfregs2 9734 dfac5lem4 10127 isf32lem9 10362 fzodisj 13673 fzodisjsn 13677 inpr0 32051 bnj1280 34344 ecin0 37537 zfregs2VD 43917 |
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