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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 3185 | . 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V) | |
2 | snex 4835 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq1 2676 | . . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V)) | |
4 | 2, 3 | mpbiri 247 | . . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V) |
5 | 4 | exlimiv 1845 | . 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V) |
6 | eleq1 2676 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons )) | |
7 | eqeq1 2614 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥})) | |
8 | 7 | exbidv 1837 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})) |
9 | df-singles 31139 | . . . . 5 ⊢ Singletons = ran Singleton | |
10 | 9 | eleq2i 2680 | . . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton) |
11 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | elrn 5287 | . . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦) |
13 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
14 | 13, 11 | brsingle 31194 | . . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥}) |
15 | 14 | exbii 1764 | . . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥}) |
16 | 10, 12, 15 | 3bitri 285 | . . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥}) |
17 | 6, 8, 16 | vtoclbg 3240 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})) |
18 | 1, 5, 17 | pm5.21nii 367 | 1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 {csn 4125 class class class wbr 4583 ran crn 5039 Singletoncsingle 31114 Singletons csingles 31115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-singleton 31138 df-singles 31139 |
This theorem is referenced by: dfsingles2 31198 snelsingles 31199 funpartlem 31219 |
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