Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfse | GIF version |
Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfse.r | ⊢ Ⅎ𝑥𝑅 |
nfse.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfse | ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 4305 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V) | |
2 | nfse.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2306 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
4 | nfse.r | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2306 | . . . . . 6 ⊢ Ⅎ𝑥𝑏 | |
6 | 3, 4, 5 | nfbr 4022 | . . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
7 | 6, 2 | nfrabxy 2644 | . . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} |
8 | 7 | nfel1 2317 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
9 | 2, 8 | nfralxy 2502 | . 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
10 | 1, 9 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 ∀wral 2442 {crab 2446 Vcvv 2721 class class class wbr 3976 Se wse 4301 |
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-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-se 4305 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |