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Mirrors > Home > ILE Home > Th. List > 0ss | GIF version |
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
0ss | ⊢ ∅ ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3290 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 610 | . 2 ⊢ (𝑥 ∈ ∅ → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3029 | 1 ⊢ ∅ ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 ⊆ wss 2999 ∅c0 3286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-in 3005 df-ss 3012 df-nul 3287 |
This theorem is referenced by: ss0b 3322 ssdifeq0 3365 sssnr 3597 ssprr 3600 uni0 3680 int0el 3718 0disj 3842 disjx0 3844 tr0 3947 0elpw 3999 fr0 4178 elnn 4420 rel0 4562 0ima 4792 fun0 5072 f0 5201 oaword1 6232 0domg 6553 nnnninf 6806 exmidfodomrlemim 6827 sum0 10780 0opn 11603 bdeq0 11758 bj-omtrans 11851 el2oss1o 11887 nninfsellemsuc 11904 |
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