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Mirrors > Home > ILE Home > Th. List > nffrfor | GIF version |
Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nffrfor.r | ⊢ Ⅎ𝑥𝑅 |
nffrfor.a | ⊢ Ⅎ𝑥𝐴 |
nffrfor.s | ⊢ Ⅎ𝑥𝑆 |
Ref | Expression |
---|---|
nffrfor | ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frfor 4303 | . 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)) | |
2 | nffrfor.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2306 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
4 | nffrfor.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2306 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
6 | 3, 4, 5 | nfbr 4022 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
7 | nffrfor.s | . . . . . . . 8 ⊢ Ⅎ𝑥𝑆 | |
8 | 7 | nfcri 2300 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆 |
9 | 6, 8 | nfim 1559 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) |
10 | 2, 9 | nfralxy 2502 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) |
11 | 7 | nfcri 2300 | . . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆 |
12 | 10, 11 | nfim 1559 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) |
13 | 2, 12 | nfralxy 2502 | . . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) |
14 | 2, 7 | nfss 3130 | . . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆 |
15 | 13, 14 | nfim 1559 | . 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆) |
16 | 1, 15 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆 |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 ∀wral 2442 ⊆ wss 3111 class class class wbr 3976 FrFor wfrfor 4299 |
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-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-frfor 4303 |
This theorem is referenced by: nffr 4321 |
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