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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsingles | Structured version Visualization version GIF version |
Description: Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.) |
Ref | Expression |
---|---|
elsingles | ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3487 | . 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V) | |
2 | vsnex 5422 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq1 2815 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V)) | |
4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V) |
5 | 4 | exlimiv 1925 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V) |
6 | eleq1 2815 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons )) | |
7 | eqeq1 2730 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥})) | |
8 | 7 | exbidv 1916 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})) |
9 | df-singles 35368 | . . . . 5 ⊢ Singletons = ran Singleton | |
10 | 9 | eleq2i 2819 | . . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton) |
11 | vex 3472 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | elrn 5887 | . . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦) |
13 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | 13, 11 | brsingle 35422 | . . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥}) |
15 | 14 | exbii 1842 | . . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥}) |
16 | 10, 12, 15 | 3bitri 297 | . . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥}) |
17 | 6, 8, 16 | vtoclbg 3539 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})) |
18 | 1, 5, 17 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 {csn 4623 class class class wbr 5141 ran crn 5670 Singletoncsingle 35343 Singletons csingles 35344 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-symdif 4237 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-1st 7974 df-2nd 7975 df-txp 35359 df-singleton 35367 df-singles 35368 |
This theorem is referenced by: dfsingles2 35426 snelsingles 35427 funpartlem 35447 |
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