![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 3491 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | eqcom 2741 | . . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
4 | 3 | exbii 1844 | . . . . 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 4640 | . . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥}) | |
8 | 6, 7 | eximii 1833 | . 2 ⊢ ∃𝑥{𝐴} = {𝑥} |
9 | elsingles 35899 | . 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥}) | |
10 | 8, 9 | mpbir 231 | 1 ⊢ {𝐴} ∈ Singletons |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 {csn 4630 Singletons csingles 35820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-symdif 4258 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-eprel 5588 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-1st 8012 df-2nd 8013 df-txp 35835 df-singleton 35843 df-singles 35844 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |