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Mirrors > Home > NFE Home > Th. List > clel3 | GIF version |
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
clel3.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
clel3 | ⊢ (A ∈ B ↔ ∃x(x = B ∧ A ∈ x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clel3.1 | . 2 ⊢ B ∈ V | |
2 | clel3g 2976 | . 2 ⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (A ∈ B ↔ ∃x(x = B ∧ A ∈ x)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 |
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: unipr 3905 nnsucelrlem1 4424 tfinnnlem1 4533 dfop2lem1 4573 setconslem2 4732 nenpw1pwlem1 6084 |
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