![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4126 | . . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵 | |
3 | 2 | biantrur 530 | . . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵)) |
4 | difsnb 4804 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | |
5 | 4 | necon3bbii 2982 | . . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵) |
6 | df-pss 3962 | . . 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 395 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 ⊆ wss 3943 ⊊ wpss 3944 {csn 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-in 3950 df-ss 3960 df-pss 3962 df-sn 4624 |
This theorem is referenced by: marypha1lem 9430 infpss 10214 ominf4 10309 mrieqv2d 17592 |
Copyright terms: Public domain | W3C validator |