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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 3924 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqss 3583 | . . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran2 956 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅) |
4 | 3 | bicomi 213 | 1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: ss0 3926 un00 3963 ssdisjOLD 3979 pw0 4283 fnsuppeq0 7210 cnfcom2lem 8481 card0 8667 kmlem5 8859 cf0 8956 fin1a2lem12 9116 mreexexlem3d 16129 efgval 17953 ppttop 20621 0nnei 20726 disjunsn 28789 isarchi 29067 filnetlem4 31546 pnonsingN 34237 osumcllem4N 34263 resnonrel 36917 ntrneicls11 37408 ntrneikb 37412 |
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