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Mirrors > Home > MPE Home > Th. List > ssdifeq0 | Structured version Visualization version GIF version |
Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
Ref | Expression |
---|---|
ssdifeq0 | ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 3930 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | ssdifin0 4158 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
3 | 1, 2 | syl5eqr 2772 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
4 | 0ss 4080 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
5 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
6 | difeq2 3830 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
7 | 5, 6 | sseq12d 3740 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) |
8 | 4, 7 | mpbiri 248 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) |
9 | 3, 8 | impbii 199 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1596 ∖ cdif 3677 ∩ cin 3679 ⊆ wss 3680 ∅c0 4023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ral 3019 df-rab 3023 df-v 3306 df-dif 3683 df-in 3687 df-ss 3694 df-nul 4024 |
This theorem is referenced by: disjdifprg 29616 measxun2 30503 measssd 30508 pmeasmono 30616 |
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