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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 3415 | . 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V) | |
2 | snex 5295 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq1 2820 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V)) | |
4 | 2, 3 | mpbiri 261 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V) |
5 | 4 | exlimiv 1936 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V) |
6 | eleq1 2820 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons )) | |
7 | eqeq1 2742 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥})) | |
8 | 7 | exbidv 1927 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})) |
9 | df-singles 33795 | . . . . 5 ⊢ Singletons = ran Singleton | |
10 | 9 | eleq2i 2824 | . . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton) |
11 | vex 3401 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | elrn 5730 | . . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦) |
13 | vex 3401 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | 13, 11 | brsingle 33849 | . . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥}) |
15 | 14 | exbii 1854 | . . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥}) |
16 | 10, 12, 15 | 3bitri 300 | . . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥}) |
17 | 6, 8, 16 | vtoclbg 3472 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})) |
18 | 1, 5, 17 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∃wex 1786 ∈ wcel 2113 Vcvv 3397 {csn 4513 class class class wbr 5027 ran crn 5520 Singletoncsingle 33770 Singletons csingles 33771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-symdif 4131 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fo 6339 df-fv 6341 df-1st 7707 df-2nd 7708 df-txp 33786 df-singleton 33794 df-singles 33795 |
This theorem is referenced by: dfsingles2 33853 snelsingles 33854 funpartlem 33874 |
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