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Mirrors > Home > NFE 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 3554 | . . 3 ⊢ ¬ x ∈ ∅ | |
2 | 1 | pm2.21i 123 | . 2 ⊢ (x ∈ ∅ → x ∈ A) |
3 | 2 | ssriv 3277 | 1 ⊢ ∅ ⊆ A |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 ⊆ wss 3257 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: ss0b 3580 0pss 3588 npss0 3589 ssdifeq0 3632 pwpw0 3855 sssn 3864 sspr 3869 sstp 3870 pwsnALT 3882 uni0 3918 int0el 3957 iotassuni 4355 0ima 5014 dmxpss 5052 dmsnopss 5067 fun0 5154 f0 5248 fvmptss 5705 fvmptss2 5725 clos10 5887 mapsspm 6021 mapsspw 6022 map0e 6023 lec0cg 6198 0lt1c 6258 frecxp 6314 |
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