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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 3482 | . 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V) | |
2 | vsnex 5425 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq1 2813 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V)) | |
4 | 2, 3 | mpbiri 257 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V) |
5 | 4 | exlimiv 1925 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V) |
6 | eleq1 2813 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons )) | |
7 | eqeq1 2729 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥})) | |
8 | 7 | exbidv 1916 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})) |
9 | df-singles 35516 | . . . . 5 ⊢ Singletons = ran Singleton | |
10 | 9 | eleq2i 2817 | . . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton) |
11 | vex 3467 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | elrn 5890 | . . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦) |
13 | vex 3467 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | 13, 11 | brsingle 35570 | . . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥}) |
15 | 14 | exbii 1842 | . . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥}) |
16 | 10, 12, 15 | 3bitri 296 | . . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥}) |
17 | 6, 8, 16 | vtoclbg 3534 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})) |
18 | 1, 5, 17 | pm5.21nii 377 | 1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3463 {csn 4624 class class class wbr 5143 ran crn 5673 Singletoncsingle 35491 Singletons csingles 35492 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-symdif 4237 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7991 df-2nd 7992 df-txp 35507 df-singleton 35515 df-singles 35516 |
This theorem is referenced by: dfsingles2 35574 snelsingles 35575 funpartlem 35595 |
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