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Mirrors > Home > NFE Home > Th. List > snprc | GIF version |
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.) |
Ref | Expression |
---|---|
snprc | ⊢ (¬ A ∈ V ↔ {A} = ∅) |
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 6312 |
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