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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjeccnvep | Structured version Visualization version GIF version |
Description: Property of the epsilon relation. (Contributed by Peter Mazsa, 27-Apr-2020.) |
Ref | Expression |
---|---|
disjeccnvep | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eccnvep 36731 | . . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) | |
2 | eccnvep 36731 | . . 3 ⊢ (𝐵 ∈ 𝑊 → [𝐵]◡ E = 𝐵) | |
3 | 1, 2 | ineqan12d 4173 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E ∩ [𝐵]◡ E ) = (𝐴 ∩ 𝐵)) |
4 | 3 | eqeq1d 2738 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3908 ∅c0 4281 E cep 5535 ◡ccnv 5631 [cec 8643 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-eprel 5536 df-xp 5638 df-rel 5639 df-cnv 5640 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ec 8647 |
This theorem is referenced by: disjecxrncnvep 36841 |
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