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Mirrors > Home > NFE Home > Th. List > abssi | GIF version |
Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) |
Ref | Expression |
---|---|
abssi.1 | ⊢ (φ → x ∈ A) |
Ref | Expression |
---|---|
abssi | ⊢ {x ∣ φ} ⊆ A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abssi.1 | . . 3 ⊢ (φ → x ∈ A) | |
2 | 1 | ss2abi 3338 | . 2 ⊢ {x ∣ φ} ⊆ {x ∣ x ∈ A} |
3 | abid2 2470 | . 2 ⊢ {x ∣ x ∈ A} = A | |
4 | 2, 3 | sseqtri 3303 | 1 ⊢ {x ∣ φ} ⊆ A |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 {cab 2339 ⊆ 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: ssab2 3350 abf 3584 intab 3956 opkabssvvk 4208 fvclss 5462 mapsspw 6022 spacssnc 6284 |
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