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Mirrors > Home > NFE Home > Th. List > nfel | GIF version |
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ ℲxA |
nfeq.2 | ⊢ ℲxB |
Ref | Expression |
---|---|
nfel | ⊢ Ⅎx A ∈ B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2349 | . 2 ⊢ (A ∈ B ↔ ∃z(z = A ∧ z ∈ B)) | |
2 | nfcv 2489 | . . . . 5 ⊢ Ⅎxz | |
3 | nfnfc.1 | . . . . 5 ⊢ ℲxA | |
4 | 2, 3 | nfeq 2496 | . . . 4 ⊢ Ⅎx z = A |
5 | nfeq.2 | . . . . 5 ⊢ ℲxB | |
6 | 5 | nfcri 2483 | . . . 4 ⊢ Ⅎx z ∈ B |
7 | 4, 6 | nfan 1824 | . . 3 ⊢ Ⅎx(z = A ∧ z ∈ B) |
8 | 7 | nfex 1843 | . 2 ⊢ Ⅎx∃z(z = A ∧ z ∈ B) |
9 | 1, 8 | nfxfr 1570 | 1 ⊢ Ⅎx A ∈ B |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 |
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-cleq 2346 df-clel 2349 df-nfc 2478 |
This theorem is referenced by: nfel1 2499 nfel2 2501 nfnel 2611 elabgf 2983 elrabf 2993 sbcel12g 3151 ffnfvf 5428 |
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