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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 2319 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2319 | . . 3 ⊢ Ⅎ𝑥∅ | |
3 | 1, 2 | cleqf 2344 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
4 | noel 3426 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | nbn 699 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
6 | 5 | albii 1470 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 3, 6 | bitr4i 187 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∀wal 1351 = wceq 1353 ∈ wcel 2148 ∅c0 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-nul 3423 |
This theorem is referenced by: notm0 3443 nel0 3444 0el 3445 rabeq0 3452 abeq0 3453 ssdif0im 3487 inssdif0im 3490 ralf0 3526 snprc 3657 uni0b 3834 disjiun 3998 0ex 4130 dm0 4841 reldm0 4845 dmsn0 5096 dmsn0el 5098 fzo0 10165 fzouzdisj 10177 |
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