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Mirrors > Home > NFE 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 | neq0 3560 | . . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A) | |
2 | df-ex 1542 | . . 3 ⊢ (∃x x ∈ A ↔ ¬ ∀x ¬ x ∈ A) | |
3 | 1, 2 | bitri 240 | . 2 ⊢ (¬ A = ∅ ↔ ¬ ∀x ¬ x ∈ A) |
4 | 3 | con4bii 288 | 1 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: 0el 3566 disj 3591 ssdif0 3609 difin0ss 3616 inssdif0 3617 ralf0 3656 addcnul1 4452 dm0 4918 dmeq0 4922 co01 5093 clos1nrel 5886 ncprc 6124 |
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