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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 5509 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)) | |
2 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
3 | nffr.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfss 3960 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴 |
5 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅ | |
6 | 4, 5 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) |
7 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑐 | |
8 | nffr.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
9 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
10 | 7, 8, 9 | nfbr 5106 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
11 | 10 | nfn 1853 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏 |
12 | 2, 11 | nfralw 3225 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
13 | 2, 12 | nfrex 3309 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
14 | 6, 13 | nfim 1893 | . . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
15 | 14 | nfal 2338 | . 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
16 | 1, 15 | nfxfr 1849 | 1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1531 Ⅎwnf 1780 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 class class class wbr 5059 Fr wfr 5506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-fr 5509 |
This theorem is referenced by: nfwe 5526 |
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