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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 3409 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 636 | . 2 ⊢ (𝑥 ∈ ∅ → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3142 | 1 ⊢ ∅ ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 ⊆ wss 3112 ∅c0 3405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-dif 3114 df-in 3118 df-ss 3125 df-nul 3406 |
This theorem is referenced by: ss0b 3444 ssdifeq0 3487 sssnr 3728 ssprr 3731 uni0 3811 int0el 3849 0disj 3974 disjx0 3976 tr0 4086 0elpw 4138 exmidsssn 4176 fr0 4324 elomssom 4577 rel0 4724 0ima 4959 fun0 5241 f0 5373 el2oss1o 6403 oaword1 6431 0domg 6795 nnnninf 7082 exmidfodomrlemim 7149 pw1on 7174 sum0 11319 prod0 11516 ennnfonelemj0 12297 ennnfonelemkh 12308 0opn 12571 baspartn 12615 0cld 12679 ntr0 12701 bdeq0 13610 bj-omtrans 13699 nninfsellemsuc 13753 |
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