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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 4261 | . 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)) | |
2 | nffrfor.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2282 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑣 | |
4 | nffrfor.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2282 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑢 | |
6 | 3, 4, 5 | nfbr 3982 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢 |
7 | nffrfor.s | . . . . . . . 8 ⊢ Ⅎ𝑥𝑆 | |
8 | 7 | nfcri 2276 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆 |
9 | 6, 8 | nfim 1552 | . . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) |
10 | 2, 9 | nfralxy 2474 | . . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) |
11 | 7 | nfcri 2276 | . . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆 |
12 | 10, 11 | nfim 1552 | . . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) |
13 | 2, 12 | nfralxy 2474 | . . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) |
14 | 2, 7 | nfss 3095 | . . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆 |
15 | 13, 14 | nfim 1552 | . 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆) |
16 | 1, 15 | nfxfr 1451 | 1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆 |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1437 ∈ wcel 1481 Ⅎwnfc 2269 ∀wral 2417 ⊆ wss 3076 class class class wbr 3937 FrFor wfrfor 4257 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-frfor 4261 |
This theorem is referenced by: nffr 4279 |
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