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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 3492 | . 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V) | |
2 | vsnex 5429 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq1 2820 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V)) | |
4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V) |
5 | 4 | exlimiv 1932 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V) |
6 | eleq1 2820 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons )) | |
7 | eqeq1 2735 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥})) | |
8 | 7 | exbidv 1923 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})) |
9 | df-singles 35306 | . . . . 5 ⊢ Singletons = ran Singleton | |
10 | 9 | eleq2i 2824 | . . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton) |
11 | vex 3477 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | elrn 5893 | . . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦) |
13 | vex 3477 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | 13, 11 | brsingle 35360 | . . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥}) |
15 | 14 | exbii 1849 | . . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥}) |
16 | 10, 12, 15 | 3bitri 297 | . . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥}) |
17 | 6, 8, 16 | vtoclbg 3544 | . 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 1540 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 {csn 4628 class class class wbr 5148 ran crn 5677 Singletoncsingle 35281 Singletons csingles 35282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-txp 35297 df-singleton 35305 df-singles 35306 |
This theorem is referenced by: dfsingles2 35364 snelsingles 35365 funpartlem 35385 |
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