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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 5637 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)) | |
| 2 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
| 3 | nffr.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nfss 3976 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴 |
| 5 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅ | |
| 6 | 4, 5 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) |
| 7 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑐 | |
| 8 | nffr.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
| 9 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
| 10 | 7, 8, 9 | nfbr 5190 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
| 11 | 10 | nfn 1857 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏 |
| 12 | 2, 11 | nfralw 3311 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
| 13 | 2, 12 | nfrexw 3313 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
| 14 | 6, 13 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
| 15 | 14 | nfal 2323 | . 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
| 16 | 1, 15 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Fr wfr 5634 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-fr 5637 |
| This theorem is referenced by: nfwe 5660 weiunfr 36468 |
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