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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 3784 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | ssdifin0 4002 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅) | |
3 | 1, 2 | syl5eqr 2658 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
4 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅) | |
5 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
6 | difeq2 3684 | . . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅)) | |
7 | 5, 6 | sseq12d 3597 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅))) |
8 | 4, 7 | mpbiri 247 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴)) |
9 | 3, 8 | impbii 198 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: disjdifprg 28770 measxun2 29600 measssd 29605 pmeasmono 29713 |
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