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Mirrors > Home > ILE Home > Th. List > ss0 | GIF version |
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.) |
Ref | Expression |
---|---|
ss0 | ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 3349 | . 2 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) | |
2 | 1 | biimpi 119 | 1 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ⊆ wss 3021 ∅c0 3310 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-dif 3023 df-in 3027 df-ss 3034 df-nul 3311 |
This theorem is referenced by: sseq0 3351 abf 3353 eq0rdv 3354 ssdisj 3366 0dif 3381 poirr2 4867 iotanul 5039 f00 5250 map0b 6511 phplem2 6676 php5dom 6686 sbthlem7 6779 casefun 6885 caseinj 6889 djufun 6904 djuinj 6906 exmidomni 6926 ixxdisj 9527 icodisj 9616 ioodisj 9617 uzdisj 9714 nn0disj 9756 fsum2dlemstep 11042 ntrcls0 12082 nninfalllemn 12786 |
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