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Theorem bnj110 32118
Description: Well-founded induction restricted to a set (𝐴 ∈ V). 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.)
Hypotheses
Ref Expression
bnj110.1 𝐴 ∈ V
bnj110.2 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
Assertion
Ref Expression
bnj110 ((𝑅 Fr 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 bnj110
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralnex 3234 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
2 sbcng 3817 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑))
32elv 3498 . . . . . . 7 ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
43bicomi 226 . . . . . 6 [𝑧 / 𝑥]𝜑[𝑧 / 𝑥] ¬ 𝜑)
54ralbii 3163 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
61, 5bitr3i 279 . . . 4 (¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
7 df-rab 3145 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
87eleq2i 2902 . . . . . 6 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
9 df-sbc 3771 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
10 sbcan 3819 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑))
11 sbcel1v 3837 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
1211anbi1i 625 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1310, 12bitri 277 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
149, 13bitr3i 279 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
158, 14bitri 277 . . . . 5 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1615simprbi 499 . . . 4 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → [𝑧 / 𝑥] ¬ 𝜑)
176, 16mprgbir 3151 . . 3 ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑
18 bnj110.1 . . . . . . . . 9 𝐴 ∈ V
1918rabex 5226 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} ∈ V
2019biantrur 533 . . . . . . 7 (𝑅 Fr 𝐴 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴))
21 rexnal 3236 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
22 rabn0 4337 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
23 ssrab2 4054 . . . . . . . . . 10 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
2423biantrur 533 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2522, 24bitr3i 279 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2621, 25bitr3i 279 . . . . . . 7 (¬ ∀𝑥𝐴 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
27 fri 5510 . . . . . . 7 ((({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
2820, 26, 27syl2anb 599 . . . . . 6 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
29 eqid 2819 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
3029bnj23 31976 . . . . . . 7 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧 → ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
31 df-ral 3141 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
3231sbcbii 3827 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ [𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
33 sbcal 3831 . . . . . . . . . 10 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
34 sbcimg 3818 . . . . . . . . . . . . 13 (𝑧 ∈ V → ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))))
3534elv 3498 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
36 vex 3496 . . . . . . . . . . . . . 14 𝑧 ∈ V
37 nfv 1908 . . . . . . . . . . . . . 14 𝑥 𝑦𝐴
3836, 37sbcgfi 3846 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑦𝐴𝑦𝐴)
39 sbcimg 3818 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑)))
4039elv 3498 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑))
41 sbcbr2g 5115 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥))
4241elv 3498 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥)
4336csbvargi 4382 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥𝑥 = 𝑧
4443breq2i 5065 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧 / 𝑥𝑥𝑦𝑅𝑧)
4542, 44bitri 277 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧)
46 nfsbc1v 3790 . . . . . . . . . . . . . . . 16 𝑥[𝑦 / 𝑥]𝜑
4736, 46sbcgfi 3846 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4845, 47imbi12i 353 . . . . . . . . . . . . . 14 (([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
4940, 48bitri 277 . . . . . . . . . . . . 13 ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5038, 49imbi12i 353 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5135, 50bitri 277 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5251albii 1813 . . . . . . . . . 10 (∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5333, 52bitri 277 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5432, 53bitri 277 . . . . . . . 8 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
55 bnj110.2 . . . . . . . . 9 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
5655sbcbii 3827 . . . . . . . 8 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
57 df-ral 3141 . . . . . . . 8 (∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5854, 56, 573bitr4i 305 . . . . . . 7 ([𝑧 / 𝑥]𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5930, 58sylibr 236 . . . . . 6 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧[𝑧 / 𝑥]𝜓)
6028, 59bnj31 31977 . . . . 5 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓)
61 nfv 1908 . . . . . . . 8 𝑧(𝜓𝜑)
62 nfsbc1v 3790 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜓
63 nfsbc1v 3790 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
6462, 63nfim 1890 . . . . . . . 8 𝑥([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)
65 sbceq1a 3781 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜓[𝑧 / 𝑥]𝜓))
66 sbceq1a 3781 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
6765, 66imbi12d 347 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝜑) ↔ ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
6861, 64, 67cbvralw 3440 . . . . . . 7 (∀𝑥𝐴 (𝜓𝜑) ↔ ∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
69 elrabi 3673 . . . . . . . . 9 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → 𝑧𝐴)
7069imim1i 63 . . . . . . . 8 ((𝑧𝐴 → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)) → (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
7170ralimi2 3155 . . . . . . 7 (∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
7268, 71sylbi 219 . . . . . 6 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
73 rexim 3239 . . . . . 6 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7472, 73syl 17 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7560, 74mpan9 509 . . . 4 (((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
7675an32s 650 . . 3 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
7717, 76mto 199 . 2 ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑)
78 iman 404 . 2 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑) ↔ ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑))
7977, 78mpbir 233 1 ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528  wcel 2107  {cab 2797  wne 3014  wral 3136  wrex 3137  {crab 3140  Vcvv 3493  [wsbc 3770  csb 3881  wss 3934  c0 4289   class class class wbr 5057   Fr wfr 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-fr 5507
This theorem is referenced by:  bnj157  32119  bnj580  32173  bnj1052  32235  bnj1030  32247  bnj1133  32249
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