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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 3482 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | eqcom 2734 | . . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
4 | 3 | exbii 1843 | . . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥) |
5 | 2, 4 | bitri 275 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥) |
6 | 1, 5 | mpbi 229 | . . 3 ⊢ ∃𝑥 𝐴 = 𝑥 |
7 | sneq 4634 | . . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥}) | |
8 | 6, 7 | eximii 1832 | . 2 ⊢ ∃𝑥{𝐴} = {𝑥} |
9 | elsingles 35437 | . 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥}) | |
10 | 8, 9 | mpbir 230 | 1 ⊢ {𝐴} ∈ Singletons |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3469 {csn 4624 Singletons csingles 35358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-symdif 4238 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7985 df-2nd 7986 df-txp 35373 df-singleton 35381 df-singles 35382 |
This theorem is referenced by: (None) |
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