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Mirrors > Home > NFE Home > Th. List > elsnc2g | GIF version |
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.) |
Ref | Expression |
---|---|
elsnc2g | ⊢ (B ∈ V → (A ∈ {B} ↔ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 3758 | . 2 ⊢ (A ∈ {B} → A = B) | |
2 | snidg 3759 | . . 3 ⊢ (B ∈ V → B ∈ {B}) | |
3 | eleq1 2413 | . . 3 ⊢ (A = B → (A ∈ {B} ↔ B ∈ {B})) | |
4 | 2, 3 | syl5ibrcom 213 | . 2 ⊢ (B ∈ V → (A = B → A ∈ {B})) |
5 | 1, 4 | impbid2 195 | 1 ⊢ (B ∈ V → (A ∈ {B} ↔ A = B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 {csn 3738 |
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-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-v 2862 df-sn 3742 |
This theorem is referenced by: elsnc2 3763 fnfreclem2 6319 fnfreclem3 6320 |
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