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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 5631 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)) | |
2 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
3 | nffr.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfss 3974 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴 |
5 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅ | |
6 | 4, 5 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) |
7 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑐 | |
8 | nffr.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
9 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
10 | 7, 8, 9 | nfbr 5195 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
11 | 10 | nfn 1860 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏 |
12 | 2, 11 | nfralw 3308 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
13 | 2, 12 | nfrexw 3310 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
14 | 6, 13 | nfim 1899 | . . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
15 | 14 | nfal 2316 | . 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
16 | 1, 15 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1539 Ⅎwnf 1785 Ⅎwnfc 2883 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 Fr wfr 5628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-fr 5631 |
This theorem is referenced by: nfwe 5652 |
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