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Theorem bnj1204 30334
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 1054 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2 ssrab2 3650 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
32a1i 11 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4 simp3 1056 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥𝐴 ¬ 𝜑)
5 rabn0 3912 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
64, 5sylibr 223 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)
7 nfrab1 3099 . . . . . . . 8 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
87nfcrii 2744 . . . . . . 7 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
98bnj1228 30333 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
101, 3, 6, 9syl3anc 1318 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11 biid 250 . . . . 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 1830 . . . . . . 7 𝑥 𝑅 FrSe 𝐴
13 nfra1 2925 . . . . . . 7 𝑥𝑥𝐴 (𝜓𝜑)
14 nfre1 2988 . . . . . . 7 𝑥𝑥𝐴 ¬ 𝜑
1512, 13, 14nf3an 1819 . . . . . 6 𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
1615nf5ri 2053 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑))
1710, 11, 16bnj1521 30175 . . . 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 2610 . . . . . 6 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
1918, 11bnj1212 30124 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥𝐴)
20 nfra1 2925 . . . . . . . 8 𝑦𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21 simp3 1056 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
2221bnj1211 30122 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23 con2b 348 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2423albii 1737 . . . . . . . . . . . . . 14 (∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2522, 24sylib 207 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
26 simp2 1055 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27 sp 2041 . . . . . . . . . . . . 13 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2825, 26, 27sylc 63 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
29 simp1 1054 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝐴)
30 nfcv 2751 . . . . . . . . . . . . . . . . . 18 𝑥𝐴
3130elrabsf 3441 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑))
32 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
33 sbcng 3443 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3534anbi2i 726 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3631, 35bitri 263 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3736notbii 309 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38 imnan 437 . . . . . . . . . . . . . . 15 ((𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3937, 38sylbb2 227 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → (𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
4039imp 444 . . . . . . . . . . . . 13 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
4140notnotrd 127 . . . . . . . . . . . 12 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → [𝑦 / 𝑥]𝜑)
4228, 29, 41syl2anc 691 . . . . . . . . . . 11 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43423expa 1257 . . . . . . . . . 10 (((𝑦𝐴𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
4443expcom 450 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦𝐴𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
4544expd 451 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
4620, 45ralrimi 2940 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
47 bnj1204.1 . . . . . . 7 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
4846, 47sylibr 223 . . . . . 6 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥𝜓)
49483ad2ant3 1077 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50 simp12 1085 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥𝐴 (𝜓𝜑))
51 simp3 1056 . . . . . . 7 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 (𝜓𝜑))
5251bnj1211 30122 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥(𝑥𝐴 → (𝜓𝜑)))
53 simp1 1054 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝑥𝐴)
54 simp2 1055 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜓)
55 sp 2041 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝜓𝜑)) → (𝑥𝐴 → (𝜓𝜑)))
5652, 53, 54, 55syl3c 64 . . . . 5 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜑)
5719, 49, 50, 56syl3anc 1318 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58 rabid 3095 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑥𝐴 ∧ ¬ 𝜑))
5958simprbi 479 . . . . 5 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60593ad2ant2 1076 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
6117, 57, 60bnj1304 30144 . . 3 ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
6261bnj1224 30126 . 2 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ¬ ∃𝑥𝐴 ¬ 𝜑)
63 dfral2 2977 . 2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
6462, 63sylibr 223 1 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031  wal 1473  wcel 1977  wne 2780  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  [wsbc 3402  wss 3540  c0 3874   class class class wbr 4583   FrSe w-bnj15 30011
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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016
This theorem is referenced by:  bnj1417  30363
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