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Theorem bnj1204 31054
Description: Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1204.1 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
Assertion
Ref Expression
bnj1204 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem bnj1204
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp1 1059 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2 ssrab2 3679 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
32a1i 11 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4 simp3 1061 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥𝐴 ¬ 𝜑)
5 rabn0 3949 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
64, 5sylibr 224 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)
7 nfrab1 3117 . . . . . . . 8 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
87nfcrii 2755 . . . . . . 7 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
98bnj1228 31053 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
101, 3, 6, 9syl3anc 1324 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11 biid 251 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥))
12 nfv 1841 . . . . . . 7 𝑥 𝑅 FrSe 𝐴
13 nfra1 2938 . . . . . . 7 𝑥𝑥𝐴 (𝜓𝜑)
14 nfre1 3002 . . . . . . 7 𝑥𝑥𝐴 ¬ 𝜑
1512, 13, 14nf3an 1829 . . . . . 6 𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
1615nf5ri 2063 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑))
1710, 11, 16bnj1521 30895 . . . 4 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥))
18 eqid 2620 . . . . . 6 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
1918, 11bnj1212 30844 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥𝐴)
20 nfra1 2938 . . . . . . . 8 𝑦𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21 simp3 1061 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
2221bnj1211 30842 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23 con2b 349 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2423albii 1745 . . . . . . . . . . . . . 14 (∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2522, 24sylib 208 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
26 simp2 1060 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27 sp 2051 . . . . . . . . . . . . 13 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2825, 26, 27sylc 65 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
29 simp1 1059 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝐴)
30 nfcv 2762 . . . . . . . . . . . . . . . . . 18 𝑥𝐴
3130elrabsf 3468 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑))
32 vex 3198 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
33 sbcng 3470 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3534anbi2i 729 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3631, 35bitri 264 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3736notbii 310 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38 imnan 438 . . . . . . . . . . . . . . 15 ((𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3937, 38sylbb2 228 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → (𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
4039imp 445 . . . . . . . . . . . . 13 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
4140notnotrd 128 . . . . . . . . . . . 12 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → [𝑦 / 𝑥]𝜑)
4228, 29, 41syl2anc 692 . . . . . . . . . . 11 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43423expa 1263 . . . . . . . . . 10 (((𝑦𝐴𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
4443expcom 451 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦𝐴𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
4544expd 452 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
4620, 45ralrimi 2954 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
47 bnj1204.1 . . . . . . 7 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
4846, 47sylibr 224 . . . . . 6 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥𝜓)
49483ad2ant3 1082 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50 simp12 1090 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥𝐴 (𝜓𝜑))
51 simp3 1061 . . . . . . 7 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 (𝜓𝜑))
5251bnj1211 30842 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥(𝑥𝐴 → (𝜓𝜑)))
53 simp1 1059 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝑥𝐴)
54 simp2 1060 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜓)
55 sp 2051 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝜓𝜑)) → (𝑥𝐴 → (𝜓𝜑)))
5652, 53, 54, 55syl3c 66 . . . . 5 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜑)
5719, 49, 50, 56syl3anc 1324 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58 rabid 3111 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑥𝐴 ∧ ¬ 𝜑))
5958simprbi 480 . . . . 5 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60593ad2ant2 1081 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
6117, 57, 60bnj1304 30864 . . 3 ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
6261bnj1224 30846 . 2 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ¬ ∃𝑥𝐴 ¬ 𝜑)
63 dfral2 2991 . 2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
6462, 63sylibr 224 1 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1479  wcel 1988  wne 2791  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  [wsbc 3429  wss 3567  c0 3907   class class class wbr 4644   FrSe w-bnj15 30732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-bnj17 30727  df-bnj14 30729  df-bnj13 30731  df-bnj15 30733  df-bnj18 30735  df-bnj19 30737
This theorem is referenced by:  bnj1417  31083
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