New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > ssriv | GIF version |
Description: Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ssriv.1 | ⊢ (x ∈ A → x ∈ B) |
Ref | Expression |
---|---|
ssriv | ⊢ A ⊆ B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3262 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
2 | ssriv.1 | . 2 ⊢ (x ∈ A → x ∈ B) | |
3 | 1, 2 | mpgbir 1550 | 1 ⊢ A ⊆ B |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-ss 3259 |
This theorem is referenced by: ssid 3290 ssv 3291 difss 3393 ssun1 3426 inss1 3475 0ss 3579 difprsnss 3846 snsspw 3877 uniin 3911 iuniin 3979 iunpwss 4055 cokrelk 4284 evenoddnnnul 4514 dmin 4913 dmcoss 4971 dminss 5041 imainss 5042 nnssnc 6147 cenc 6181 |
Copyright terms: Public domain | W3C validator |