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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 4355 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
2 | df-ral 3111 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitri 278 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-dif 3884 df-in 3888 df-nul 4244 |
This theorem is referenced by: reldisj 4359 reldisjOLD 4360 disj3 4361 undif4 4374 disjsn 4607 funun 6370 zfregs2 9159 dfac5lem4 9537 isf32lem9 9772 fzodisj 13066 fzodisjsn 13070 inpr0 30304 bnj1280 32402 ecin0 35766 zfregs2VD 41547 |
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