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Mirrors > Home > NFE Home > Th. List > clel3g | GIF version |
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |
Ref | Expression |
---|---|
clel3g | ⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2414 | . . 3 ⊢ (x = B → (A ∈ x ↔ A ∈ B)) | |
2 | 1 | ceqsexgv 2971 | . 2 ⊢ (B ∈ V → (∃x(x = B ∧ A ∈ x) ↔ A ∈ B)) |
3 | 2 | bicomd 192 | 1 ⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 |
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 2478 df-v 2861 |
This theorem is referenced by: clel3 2977 dfiun2g 3999 |
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