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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 2175 | . . 3 ⊢ ℲxA | |
2 | nfcv 2175 | . . 3 ⊢ Ⅎx∅ | |
3 | 1, 2 | cleqf 2198 | . 2 ⊢ (A = ∅ ↔ ∀x(x ∈ A ↔ x ∈ ∅)) |
4 | noel 3222 | . . . 4 ⊢ ¬ x ∈ ∅ | |
5 | 4 | nbn 614 | . . 3 ⊢ (¬ x ∈ A ↔ (x ∈ A ↔ x ∈ ∅)) |
6 | 5 | albii 1356 | . 2 ⊢ (∀x ¬ x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ ∅)) |
7 | 3, 6 | bitr4i 176 | 1 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∅c0 3218 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-nul 3219 |
This theorem is referenced by: 0el 3235 rabeq0 3241 abeq0 3242 ssdif0im 3280 inssdif0im 3285 ralf0 3318 snprc 3426 uni0b 3596 0ex 3875 dm0 4492 reldm0 4496 dmsn0 4731 dmsn0el 4733 fzo0 8794 fzouzdisj 8806 |
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