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Mirrors > Home > MPE Home > Th. List > nffr | Structured version Visualization version 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 |
---|---|
nffr.r | ⊢ Ⅎ𝑥𝑅 |
nffr.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nffr | ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fr 5362 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)) | |
2 | nfcv 2926 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
3 | nffr.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfss 3845 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴 |
5 | nfv 1873 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅ | |
6 | 4, 5 | nfan 1862 | . . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) |
7 | nfcv 2926 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑐 | |
8 | nffr.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
9 | nfcv 2926 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
10 | 7, 8, 9 | nfbr 4972 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
11 | 10 | nfn 1819 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏 |
12 | 2, 11 | nfral 3168 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
13 | 2, 12 | nfrex 3247 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
14 | 6, 13 | nfim 1859 | . . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
15 | 14 | nfal 2263 | . 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
16 | 1, 15 | nfxfr 1815 | 1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∀wal 1505 Ⅎwnf 1746 Ⅎwnfc 2910 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 ⊆ wss 3823 ∅c0 4172 class class class wbr 4925 Fr wfr 5359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-fr 5362 |
This theorem is referenced by: nfwe 5379 |
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