Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE 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 | velsn 3392 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | exbii 1496 | . . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴) |
3 | 2 | notbii 594 | . 2 ⊢ (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
4 | eq0 3239 | . . 3 ⊢ ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴}) | |
5 | alnex 1388 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴}) | |
6 | 4, 5 | bitri 173 | . 2 ⊢ ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴}) |
7 | isset 2561 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
8 | 7 | notbii 594 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
9 | 3, 6, 8 | 3bitr4ri 202 | 1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-nul 3225 df-sn 3381 |
This theorem is referenced by: prprc1 3478 prprc 3480 snexprc 3938 sucprc 4149 snnen2oprc 6323 |
Copyright terms: Public domain | W3C validator |