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Mirrors > Home > MPE Home > Th. List > snprc | Structured version Visualization version 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, 21-Jun-1993.) |
Ref | Expression |
---|---|
snprc | ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 4646 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | exbii 1844 | . . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴) |
3 | neq0 4357 | . . 3 ⊢ (¬ {𝐴} = ∅ ↔ ∃𝑥 𝑥 ∈ {𝐴}) | |
4 | isset 3491 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
5 | 2, 3, 4 | 3bitr4i 303 | . 2 ⊢ (¬ {𝐴} = ∅ ↔ 𝐴 ∈ V) |
6 | 5 | con1bii 356 | 1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 df-nul 4339 df-sn 4631 |
This theorem is referenced by: snnzb 4722 rmosn 4723 rabsnif 4727 prprc1 4769 prprc 4771 preqsnd 4863 unisn2 5317 eqsnuniex 5366 snexALT 5388 snex 5441 posn 5773 frsn 5775 relsnb 5814 relimasn 6104 elimasni 6111 inisegn0 6118 dmsnsnsn 6241 predprc 6360 sucprc 6461 dffv3 6902 fconst5 7225 ordsuci 7827 1stval 8014 2ndval 8015 ecexr 8748 snfi 9081 snfiOLD 9082 domunsn 9165 snnen2oOLD 9261 hashrabrsn 14407 hashrabsn01 14408 hashrabsn1 14409 elprchashprn2 14431 hashsn01 14451 hash2pwpr 14511 snsymgefmndeq 19426 efgrelexlema 19781 usgr1v 29287 1conngr 30222 frgr1v 30299 n0lplig 30511 unidifsnne 32561 eldm3 35740 opelco3 35755 fvsingle 35901 unisnif 35906 funpartlem 35923 bj-sngltag 36965 bj-snex 37017 bj-restsnid 37069 bj-snmooreb 37096 wopprc 43018 safesnsupfidom1o 43406 sn1dom 43515 uneqsn 44014 vsn 48659 mofsn2 48674 |
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