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Mirrors > Home > MPE Home > Th. List > ss0b | Structured version Visualization version 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 4392 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqss 3993 | . . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran2 708 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅) |
4 | 3 | bicomi 223 | 1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ⊆ wss 3944 ∅c0 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3947 df-in 3951 df-ss 3961 df-nul 4319 |
This theorem is referenced by: ss0 4394 un00 4438 pw0 4808 al0ssb 5301 fnsuppeq0 8159 cnfcom2lem 9678 card0 9935 kmlem5 10131 cf0 10228 fin1a2lem12 10388 mreexexlem3d 17572 efgval 19549 ppttop 22439 0nnei 22545 disjunsn 31690 isarchi 32199 filnetlem4 35070 bj-pw0ALT 35734 coss0 37154 pnonsingN 38609 osumcllem4N 38635 resnonrel 42114 ntrneicls11 42612 ntrneikb 42616 sprsymrelfvlem 45930 |
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