![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssundif | Structured version Visualization version GIF version |
Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.) |
Ref | Expression |
---|---|
ssundif | ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.6 1001 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) | |
2 | eldif 3959 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | imbi1i 350 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
4 | elun 4149 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 4 | imbi2i 336 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
6 | 1, 3, 5 | 3bitr4ri 304 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
7 | 6 | albii 1822 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
8 | dfss2 3969 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶))) | |
9 | dfss2 3969 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) | |
10 | 7, 8, 9 | 3bitr4i 303 | 1 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∀wal 1540 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 |
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-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 |
This theorem is referenced by: difcom 4489 uneqdifeq 4493 ssunsn2 4831 elpwun 7756 soex 7912 ressuppssdif 8170 ssfi 9173 frfi 9288 cantnfp1lem3 9675 dfacfin7 10394 zornn0g 10500 fpwwe2lem12 10637 hashbclem 14411 incexclem 15782 ramub1lem1 16959 lpcls 22868 cmpcld 22906 alexsubALTlem3 23553 restmetu 24079 uniiccdif 25095 abelthlem2 25944 abelthlem3 25945 pmtrcnelor 32252 imadifss 36463 frege124d 42512 |
Copyright terms: Public domain | W3C validator |