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Mirrors > Home > MPE Home > Th. List > difsnpss | Structured version Visualization version GIF version |
Description: (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if and only if 𝐴 is a member of 𝐵. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difsnpss | ⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 315 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ¬ ¬ 𝐴 ∈ 𝐵) | |
2 | difss 4092 | . . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵 | |
3 | 2 | biantrur 532 | . . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵)) |
4 | difsnb 4767 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | |
5 | 4 | necon3bbii 2988 | . . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | df-pss 3930 | . . 3 ⊢ ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵)) | |
7 | 3, 5, 6 | 3bitr4i 303 | . 2 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) |
8 | 1, 7 | bitri 275 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ≠ wne 2940 ∖ cdif 3908 ⊆ wss 3911 ⊊ wpss 3912 {csn 4587 |
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-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-pss 3930 df-sn 4588 |
This theorem is referenced by: marypha1lem 9374 infpss 10158 ominf4 10253 mrieqv2d 17524 |
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