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Mirrors > Home > MPE Home > Th. List > reldisj | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid ax-12 2172. (Revised by Gino Giotto, 28-Jun-2024.) |
Ref | Expression |
---|---|
reldisj | ⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3969 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
2 | eleq1w 2817 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
3 | eleq1w 2817 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶)) | |
4 | 2, 3 | imbi12d 345 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐶))) |
5 | 4 | spw 2038 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) |
6 | pm5.44 544 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))) | |
7 | eldif 3959 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
8 | 7 | imbi2i 336 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))) |
9 | 6, 8 | bitr4di 289 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))) |
11 | 1, 10 | sylbi 216 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))) |
12 | 11 | albidv 1924 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))) |
13 | disj1 4451 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
14 | dfss2 3969 | . 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵))) | |
15 | 12, 13, 14 | 3bitr4g 314 | 1 ⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 |
This theorem is referenced by: disj2 4458 ssdifsn 4792 oacomf1olem 8564 domdifsn 9054 elfiun 9425 cantnfp1lem3 9675 ssxr 11283 structcnvcnv 17086 fidomndrng 20926 elcls 22577 ist1-2 22851 nrmsep2 22860 nrmsep 22861 isnrm3 22863 isreg2 22881 hauscmplem 22910 connsub 22925 iunconnlem 22931 llycmpkgen2 23054 hausdiag 23149 trfil3 23392 isufil2 23412 filufint 23424 blcld 24014 i1fima2 25196 i1fd 25198 nbgrssvwo2 28619 pliguhgr 29739 symgcom2 32245 inunissunidif 36256 poimirlem15 36503 itg2addnclem2 36540 ntrk0kbimka 42790 ntrneicls11 42841 gneispace 42885 opndisj 47535 seposep 47558 |
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