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Theorem snprc 3788

Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

**Assertion**
Ref	Expression
snprc	⊢ (¬ A ∈ V ↔ {A} = ∅)

Proof of Theorem snprc Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsn 3748	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
2	1	exbii 1582	. . 3 ⊢ (∃x x ∈ {A} ↔ ∃x x = A)
3		neq0 3560	. . 3 ⊢ (¬ {A} = ∅ ↔ ∃x x ∈ {A})
4		isset 2863	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
5	2, 3, 4	3bitr4i 268	. 2 ⊢ (¬ {A} = ∅ ↔ A ∈ V)
6	5	con1bii 321	1 ⊢ (¬ A ∈ V ↔ {A} = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 176 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∅c0 3550 {csn 3737

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-sn 3741

This theorem is referenced by: snex 4111 prprc2 4122 0nel1c 4159 snfi 4431 imasn 5018 dmsnopss 5067 fconst5 5455 ecexr 5950 frecxp 6314

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