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Theorem bnj1204 35004
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 1135 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2 ssrab2 4089 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
32a1i 11 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4 simp3 1137 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥𝐴 ¬ 𝜑)
5 rabn0 4394 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
64, 5sylibr 234 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)
7 nfrab1 3453 . . . . . . . 8 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
87nfcrii 2897 . . . . . . 7 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑})
98bnj1228 35003 . . . . . 6 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
101, 3, 6, 9syl3anc 1370 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11 biid 261 . . . . 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 1911 . . . . . . 7 𝑥 𝑅 FrSe 𝐴
13 nfra1 3281 . . . . . . 7 𝑥𝑥𝐴 (𝜓𝜑)
14 nfre1 3282 . . . . . . 7 𝑥𝑥𝐴 ¬ 𝜑
1512, 13, 14nf3an 1898 . . . . . 6 𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
1615nf5ri 2192 . . . . 5 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑))
1710, 11, 16bnj1521 34843 . . . 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 2734 . . . . . 6 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
1918, 11bnj1212 34791 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥𝐴)
20 nfra1 3281 . . . . . . . 8 𝑦𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21 simp3 1137 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
2221bnj1211 34789 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23 con2b 359 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2423albii 1815 . . . . . . . . . . . . . 14 (∀𝑦(𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
2522, 24sylib 218 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑}))
26 simp2 1136 . . . . . . . . . . . . 13 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27 sp 2180 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝐴)
30 nfcv 2902 . . . . . . . . . . . . . . . . . 18 𝑥𝐴
3130elrabsf 3839 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑))
32 vex 3481 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
33 sbcng 3841 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3534anbi2i 623 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴[𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3631, 35bitri 275 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3736notbii 320 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38 imnan 399 . . . . . . . . . . . . . . 15 ((𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
3937, 38sylbb2 238 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → (𝑦𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
4039imp 406 . . . . . . . . . . . . 13 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
4140notnotrd 133 . . . . . . . . . . . 12 ((¬ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ 𝑦𝐴) → [𝑦 / 𝑥]𝜑)
4228, 29, 41syl2anc 584 . . . . . . . . . . 11 ((𝑦𝐴𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43423expa 1117 . . . . . . . . . 10 (((𝑦𝐴𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
4443expcom 413 . . . . . . . . 9 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦𝐴𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
4544expd 415 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
4620, 45ralrimi 3254 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
47 bnj1204.1 . . . . . . 7 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
4846, 47sylibr 234 . . . . . 6 (∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥𝜓)
49483ad2ant3 1134 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50 simp12 1203 . . . . 5 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥𝐴 (𝜓𝜑))
51 simp3 1137 . . . . . . 7 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 (𝜓𝜑))
5251bnj1211 34789 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥(𝑥𝐴 → (𝜓𝜑)))
53 simp1 1135 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝑥𝐴)
54 simp2 1136 . . . . . 6 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜓)
55 sp 2180 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝜓𝜑)) → (𝑥𝐴 → (𝜓𝜑)))
5652, 53, 54, 55syl3c 66 . . . . 5 ((𝑥𝐴𝜓 ∧ ∀𝑥𝐴 (𝜓𝜑)) → 𝜑)
5719, 49, 50, 56syl3anc 1370 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58 rabid 3454 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑥𝐴 ∧ ¬ 𝜑))
5958simprbi 496 . . . . 5 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60593ad2ant2 1133 . . . 4 (((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
6117, 57, 60bnj1304 34811 . . 3 ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 ¬ 𝜑)
6261bnj1224 34793 . 2 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ¬ ∃𝑥𝐴 ¬ 𝜑)
63 dfral2 3096 . 2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
6462, 63sylibr 234 1 ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1534  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  [wsbc 3790  wss 3962  c0 4338   class class class wbr 5147   FrSe w-bnj15 34684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-bnj17 34679  df-bnj14 34681  df-bnj13 34683  df-bnj15 34685  df-bnj18 34687  df-bnj19 34689
This theorem is referenced by:  bnj1417  35033
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