New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > ss0b | GIF version |
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
ss0b | ⊢ (A ⊆ ∅ ↔ A = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3579 | . . 3 ⊢ ∅ ⊆ A | |
2 | eqss 3287 | . . 3 ⊢ (A = ∅ ↔ (A ⊆ ∅ ∧ ∅ ⊆ A)) | |
3 | 1, 2 | mpbiran2 885 | . 2 ⊢ (A = ∅ ↔ A ⊆ ∅) |
4 | 3 | bicomi 193 | 1 ⊢ (A ⊆ ∅ ↔ A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ⊆ wss 3257 ∅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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: ss0 3581 un00 3586 ssdisj 3600 pw0 4160 ssfin 4470 |
Copyright terms: Public domain | W3C validator |