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Mirrors > Home > ILE 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 2036 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
3 | nfnfc.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfeq 2185 | . . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴 |
5 | nfeq.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfcri 2172 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
7 | 4, 6 | nfan 1457 | . . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵) |
8 | 7 | nfex 1528 | . 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵) |
9 | 1, 8 | nfxfr 1363 | 1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 ∈ wcel 1393 Ⅎwnfc 2165 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 |
This theorem is referenced by: nfel1 2188 nfel2 2190 nfnel 2304 elabgf 2685 elrabf 2696 sbcel12g 2865 nfdisjv 3757 rabxfrd 4201 ffnfvf 5324 elabgft1 9917 elabgf2 9919 bj-rspgt 9925 |
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