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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 2336 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2336 | . . 3 ⊢ Ⅎ𝑥∅ | |
3 | 1, 2 | cleqf 2361 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
4 | noel 3450 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | nbn 700 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
6 | 5 | albii 1481 | . 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 1362 = wceq 1364 ∈ wcel 2164 ∅c0 3446 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-dif 3155 df-nul 3447 |
This theorem is referenced by: notm0 3467 nel0 3468 0el 3469 rabeq0 3476 abeq0 3477 ssdif0im 3511 inssdif0im 3514 ralf0 3549 snprc 3683 uni0b 3860 disjiun 4024 0ex 4156 dm0 4870 reldm0 4874 dmsn0 5125 dmsn0el 5127 fzo0 10225 fzouzdisj 10237 |
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