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Mirrors > Home > NFE Home > Th. List > snssd | GIF version |
Description: The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
snssd.1 | ⊢ (φ → A ∈ B) |
Ref | Expression |
---|---|
snssd | ⊢ (φ → {A} ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssd.1 | . 2 ⊢ (φ → A ∈ B) | |
2 | snssg 3844 | . . 3 ⊢ (A ∈ B → (A ∈ B ↔ {A} ⊆ B)) | |
3 | 1, 2 | syl 15 | . 2 ⊢ (φ → (A ∈ B ↔ {A} ⊆ B)) |
4 | 1, 3 | mpbid 201 | 1 ⊢ (φ → {A} ⊆ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∈ wcel 1710 ⊆ wss 3257 {csn 3737 |
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 df-sn 3741 |
This theorem is referenced by: nchoicelem6 6294 frecsuc 6322 |
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