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Theorem bnj1204 31418
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 1130 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2 ssrab2 3836 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
32a1i 11 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4 simp3 1132 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥𝐴 ¬ 𝜑)
5 rabn0 4104 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
64, 5sylibr 224 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)
7 nfrab1 3271 . . . . . . . 8 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
87nfcrii 2906 . . . . . . 7 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
98bnj1228 31417 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
101, 3, 6, 9syl3anc 1476 . . . . 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 1995 . . . . . . 7 𝑥 𝑅 FrSe 𝐴
13 nfra1 3090 . . . . . . 7 𝑥𝑥𝐴 (𝜓𝜑)
14 nfre1 3153 . . . . . . 7 𝑥𝑥𝐴 ¬ 𝜑
1512, 13, 14nf3an 1983 . . . . . 6 𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
1615nf5ri 2219 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑))
1710, 11, 16bnj1521 31259 . . . 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 2771 . . . . . 6 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
1918, 11bnj1212 31208 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥𝐴)
20 nfra1 3090 . . . . . . . 8 𝑦𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21 simp3 1132 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
2221bnj1211 31206 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23 con2b 348 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2423albii 1895 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27 sp 2207 . . . . . . . . . . . . 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 1130 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝐴)
30 nfcv 2913 . . . . . . . . . . . . . . . . . 18 𝑥𝐴
3130elrabsf 3626 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑))
32 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
33 sbcng 3628 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3534anbi2i 609 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3631, 35bitri 264 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3736notbii 309 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38 imnan 386 . . . . . . . . . . . . . . 15 ((𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3937, 38sylbb2 228 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → (𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
4039imp 393 . . . . . . . . . . . . 13 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
4140notnotrd 130 . . . . . . . . . . . 12 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → [𝑦 / 𝑥]𝜑)
4228, 29, 41syl2anc 573 . . . . . . . . . . 11 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43423expa 1111 . . . . . . . . . 10 (((𝑦𝐴𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
4443expcom 398 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦𝐴𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
4544expd 400 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
4620, 45ralrimi 3106 . . . . . . 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 1129 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50 simp12 1246 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥𝐴 (𝜓𝜑))
51 simp3 1132 . . . . . . 7 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 (𝜓𝜑))
5251bnj1211 31206 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥(𝑥𝐴 → (𝜓𝜑)))
53 simp1 1130 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝑥𝐴)
54 simp2 1131 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜓)
55 sp 2207 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝜓𝜑)) → (𝑥𝐴 → (𝜓𝜑)))
5652, 53, 54, 55syl3c 66 . . . . 5 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜑)
5719, 49, 50, 56syl3anc 1476 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58 rabid 3264 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑥𝐴 ∧ ¬ 𝜑))
5958simprbi 484 . . . . 5 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60593ad2ant2 1128 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
6117, 57, 60bnj1304 31228 . . 3 ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
6261bnj1224 31210 . 2 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ¬ ∃𝑥𝐴 ¬ 𝜑)
63 dfral2 3142 . 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 382  w3a 1071  wal 1629  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  [wsbc 3587  wss 3723  c0 4063   class class class wbr 4786   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-1o 7713  df-bnj17 31093  df-bnj14 31095  df-bnj13 31097  df-bnj15 31099  df-bnj18 31101  df-bnj19 31103
This theorem is referenced by:  bnj1417  31447
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