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Theorem bnj1204 30179
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 1053 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2 ssrab2 3554 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
32a1i 11 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4 simp3 1055 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥𝐴 ¬ 𝜑)
5 rabn0 3815 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
64, 5sylibr 222 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)
7 nfrab1 3003 . . . . . . . 8 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
87nfcrii 2648 . . . . . . 7 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
98bnj1228 30178 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
101, 3, 6, 9syl3anc 1317 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11 biid 249 . . . . 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 1796 . . . . . . 7 𝑥 𝑅 FrSe 𝐴
13 nfra1 2829 . . . . . . 7 𝑥𝑥𝐴 (𝜓𝜑)
14 nfre1 2892 . . . . . . 7 𝑥𝑥𝐴 ¬ 𝜑
1512, 13, 14nf3an 2061 . . . . . 6 𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
1615nfri 2005 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑))
1710, 11, 16bnj1521 30020 . . . 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 2514 . . . . . 6 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
1918, 11bnj1212 29969 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥𝐴)
20 nfra1 2829 . . . . . . . 8 𝑦𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21 simp3 1055 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
2221bnj1211 29967 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23 con2b 347 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2423albii 1722 . . . . . . . . . . . . . 14 (∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2522, 24sylib 206 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
26 simp2 1054 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27 sp 1990 . . . . . . . . . . . . 13 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2825, 26, 27sylc 62 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
29 simp1 1053 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝐴)
30 nfcv 2655 . . . . . . . . . . . . . . . . . 18 𝑥𝐴
3130elrabsf 3345 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑))
32 vex 3080 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
33 sbcng 3347 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3534anbi2i 725 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3631, 35bitri 262 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3736notbii 308 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38 imnan 436 . . . . . . . . . . . . . . 15 ((𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3937, 38sylbb2 226 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → (𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
4039imp 443 . . . . . . . . . . . . 13 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
4140notnotrd 126 . . . . . . . . . . . 12 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → [𝑦 / 𝑥]𝜑)
4228, 29, 41syl2anc 690 . . . . . . . . . . 11 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43423expa 1256 . . . . . . . . . 10 (((𝑦𝐴𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
4443expcom 449 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦𝐴𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
4544expd 450 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
4620, 45ralrimi 2844 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
47 bnj1204.1 . . . . . . 7 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
4846, 47sylibr 222 . . . . . 6 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥𝜓)
49483ad2ant3 1076 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50 simp12 1084 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥𝐴 (𝜓𝜑))
51 simp3 1055 . . . . . . 7 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 (𝜓𝜑))
5251bnj1211 29967 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥(𝑥𝐴 → (𝜓𝜑)))
53 simp1 1053 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝑥𝐴)
54 simp2 1054 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜓)
55 sp 1990 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝜓𝜑)) → (𝑥𝐴 → (𝜓𝜑)))
5652, 53, 54, 55syl3c 63 . . . . 5 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜑)
5719, 49, 50, 56syl3anc 1317 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58 rabid 2999 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑥𝐴 ∧ ¬ 𝜑))
5958simprbi 478 . . . . 5 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60593ad2ant2 1075 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
6117, 57, 60bnj1304 29989 . . 3 ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
6261bnj1224 29971 . 2 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ¬ ∃𝑥𝐴 ¬ 𝜑)
63 dfral2 2881 . 2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
6462, 63sylibr 222 1 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030  wal 1472  wcel 1938  wne 2684  wral 2800  wrex 2801  {crab 2804  Vcvv 3077  [wsbc 3306  wss 3444  c0 3777   class class class wbr 4481   FrSe w-bnj15 29856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-reg 8256  ax-inf2 8297
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-om 6834  df-1o 7323  df-bnj17 29851  df-bnj14 29853  df-bnj13 29855  df-bnj15 29857  df-bnj18 29859  df-bnj19 29861
This theorem is referenced by:  bnj1417  30208
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