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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 4997 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)) | |
2 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
3 | nffr.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfss 3561 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴 |
5 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅ | |
6 | 4, 5 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) |
7 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑐 | |
8 | nffr.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
9 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑏 | |
10 | 7, 8, 9 | nfbr 4629 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏 |
11 | 10 | nfn 1768 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏 |
12 | 2, 11 | nfral 2929 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
13 | 2, 12 | nfrex 2990 | . . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏 |
14 | 6, 13 | nfim 1813 | . . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
15 | 14 | nfal 2139 | . 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏) |
16 | 1, 15 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 Ⅎwnf 1699 Ⅎwnfc 2738 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 Fr wfr 4994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-fr 4997 |
This theorem is referenced by: nfwe 5014 |
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