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Mirrors > Home > NFE Home > Th. List > snssg | GIF version |
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) |
Ref | Expression |
---|---|
snssg | ⊢ (A ∈ V → (A ∈ B ↔ {A} ⊆ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2413 | . 2 ⊢ (x = A → (x ∈ B ↔ A ∈ B)) | |
2 | sneq 3744 | . . 3 ⊢ (x = A → {x} = {A}) | |
3 | 2 | sseq1d 3298 | . 2 ⊢ (x = A → ({x} ⊆ B ↔ {A} ⊆ B)) |
4 | vex 2862 | . . 3 ⊢ x ∈ V | |
5 | 4 | snss 3838 | . 2 ⊢ (x ∈ B ↔ {x} ⊆ B) |
6 | 1, 3, 5 | vtoclbg 2915 | 1 ⊢ (A ∈ V → (A ∈ B ↔ {A} ⊆ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 {csn 3737 |
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-ss 3259 df-sn 3741 |
This theorem is referenced by: snssi 3852 snssd 3853 prssg 3862 snelpwg 4114 elssetkg 4269 nnadjoinpw 4521 spacid 6283 |
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