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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 3506 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | eqcom 2828 | . . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
4 | 3 | exbii 1844 | . . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥) |
5 | 2, 4 | bitri 277 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥) |
6 | 1, 5 | mpbi 232 | . . 3 ⊢ ∃𝑥 𝐴 = 𝑥 |
7 | sneq 4570 | . . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥}) | |
8 | 6, 7 | eximii 1833 | . 2 ⊢ ∃𝑥{𝐴} = {𝑥} |
9 | elsingles 33374 | . 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥}) | |
10 | 8, 9 | mpbir 233 | 1 ⊢ {𝐴} ∈ Singletons |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 {csn 4560 Singletons csingles 33295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-eprel 5459 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7683 df-2nd 7684 df-txp 33310 df-singleton 33318 df-singles 33319 |
This theorem is referenced by: (None) |
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