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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 3486 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | eqcom 2738 | . . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
4 | 3 | exbii 1849 | . . . . 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 4638 | . . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥}) | |
8 | 6, 7 | eximii 1838 | . 2 ⊢ ∃𝑥{𝐴} = {𝑥} |
9 | elsingles 35361 | . 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥}) | |
10 | 8, 9 | mpbir 230 | 1 ⊢ {𝐴} ∈ Singletons |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 {csn 4628 Singletons csingles 35282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-txp 35297 df-singleton 35305 df-singles 35306 |
This theorem is referenced by: (None) |
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