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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsingles2 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
dfsingles2 | ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsingles 35423 | . 2 ⊢ (𝑥 ∈ Singletons ↔ ∃𝑦 𝑥 = {𝑦}) | |
2 | 1 | eqabi 2863 | 1 ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 {cab 2703 {csn 4623 Singletons csingles 35344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-symdif 4237 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-1st 7974 df-2nd 7975 df-txp 35359 df-singleton 35367 df-singles 35368 |
This theorem is referenced by: dfiota3 35428 |
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