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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 2160 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nfcv 2306 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
3 | nfnfc.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfeq 2314 | . . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴 |
5 | nfeq.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
7 | 4, 6 | nfan 1552 | . . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵) |
8 | 7 | nfex 1624 | . 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵) |
9 | 1, 8 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 Ⅎwnf 1447 ∃wex 1479 ∈ wcel 2135 Ⅎwnfc 2293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-nfc 2295 |
This theorem is referenced by: nfel1 2317 nfel2 2319 nfnel 2436 elabgf 2863 elrabf 2875 sbcel12g 3055 nfdisjv 3965 rabxfrd 4441 ffnfvf 5638 mptelixpg 6691 elabgft1 13494 elabgf2 13496 bj-rspgt 13502 |
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