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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 35571 | . 2 ⊢ (𝑥 ∈ Singletons ↔ ∃𝑦 𝑥 = {𝑦}) | |
2 | 1 | eqabi 2861 | 1 ⊢ Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 {cab 2702 {csn 4624 Singletons csingles 35492 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-symdif 4237 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7991 df-2nd 7992 df-txp 35507 df-singleton 35515 df-singles 35516 |
This theorem is referenced by: dfiota3 35576 |
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