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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 | pm5.6 995 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) | |
2 | eldif 3943 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | imbi1i 351 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)) |
4 | elun 4122 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) | |
5 | 4 | imbi2i 337 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))) |
6 | 1, 3, 5 | 3bitr4ri 305 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
7 | 6 | albii 1811 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) |
8 | dfss2 3952 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶))) | |
9 | dfss2 3952 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶)) | |
10 | 7, 8, 9 | 3bitr4i 304 | 1 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∀wal 1526 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 |
This theorem is referenced by: difcom 4430 uneqdifeq 4434 ssunsn2 4752 elpwun 7480 soex 7615 ressuppssdif 7840 frfi 8751 cantnfp1lem3 9131 dfacfin7 9809 zornn0g 9915 fpwwe2lem13 10052 hashbclem 13798 incexclem 15179 ramub1lem1 16350 lpcls 21900 cmpcld 21938 alexsubALTlem3 22585 restmetu 23107 uniiccdif 24106 abelthlem2 24947 abelthlem3 24948 pmtrcnelor 30662 imadifss 34748 frege124d 39984 |
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