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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 316 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ¬ ¬ 𝐴 ∈ 𝐵) | |
2 | difss 4105 | . . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵 | |
3 | 2 | biantrur 531 | . . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵)) |
4 | difsnb 4731 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | |
5 | 4 | necon3bbii 3060 | . . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | df-pss 3951 | . . 3 ⊢ ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵)) | |
7 | 3, 5, 6 | 3bitr4i 304 | . 2 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) |
8 | 1, 7 | bitri 276 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 ⊆ wss 3933 ⊊ wpss 3934 {csn 4557 |
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-ne 3014 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-pss 3951 df-sn 4558 |
This theorem is referenced by: marypha1lem 8885 infpss 9627 ominf4 9722 mrieqv2d 16898 |
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