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Mirrors > Home > ILE Home > Th. List > eq0 | GIF version |
Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) |
Ref | Expression |
---|---|
eq0 | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2219 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2219 | . . 3 ⊢ Ⅎ𝑥∅ | |
3 | 1, 2 | cleqf 2242 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
4 | noel 3255 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | nbn 647 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
6 | 5 | albii 1399 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 3, 6 | bitr4i 185 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 ∅c0 3251 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-nul 3252 |
This theorem is referenced by: 0el 3268 rabeq0 3274 abeq0 3275 ssdif0im 3308 inssdif0im 3311 ralf0 3344 snprc 3457 uni0b 3626 0ex 3905 dm0 4567 reldm0 4571 dmsn0 4808 dmsn0el 4810 fzo0 9177 fzouzdisj 9189 |
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