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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snelsingles | Structured version Visualization version GIF version | ||
| Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.) |
| Ref | Expression |
|---|---|
| snelsingles.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| snelsingles | ⊢ {𝐴} ∈ Singletons |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snelsingles.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 2 | isset 3494 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
| 3 | eqcom 2744 | . . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
| 4 | 3 | exbii 1848 | . . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥) |
| 5 | 2, 4 | bitri 275 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥) |
| 6 | 1, 5 | mpbi 230 | . . 3 ⊢ ∃𝑥 𝐴 = 𝑥 |
| 7 | sneq 4636 | . . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥}) | |
| 8 | 6, 7 | eximii 1837 | . 2 ⊢ ∃𝑥{𝐴} = {𝑥} |
| 9 | elsingles 35919 | . 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥}) | |
| 10 | 8, 9 | mpbir 231 | 1 ⊢ {𝐴} ∈ Singletons |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {csn 4626 Singletons csingles 35840 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-singleton 35863 df-singles 35864 |
| This theorem is referenced by: (None) |
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