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Theorem bnj110 31054
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 3021 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
2 vex 3234 . . . . . . . 8 𝑧 ∈ V
3 sbcng 3509 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑))
42, 3ax-mp 5 . . . . . . 7 ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
54bicomi 214 . . . . . 6 [𝑧 / 𝑥]𝜑[𝑧 / 𝑥] ¬ 𝜑)
65ralbii 3009 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
71, 6bitr3i 266 . . . 4 (¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
8 df-rab 2950 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
98eleq2i 2722 . . . . . 6 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
10 df-sbc 3469 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
11 sbcan 3511 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑))
12 sbcel1v 3528 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
1312anbi1i 731 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1411, 13bitri 264 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1510, 14bitr3i 266 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
169, 15bitri 264 . . . . 5 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1716simprbi 479 . . . 4 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → [𝑧 / 𝑥] ¬ 𝜑)
187, 17mprgbir 2956 . . 3 ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑
19 bnj110.1 . . . . . . . . 9 𝐴 ∈ V
2019rabex 4845 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} ∈ V
2120biantrur 526 . . . . . . 7 (𝑅 Fr 𝐴 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴))
22 rexnal 3024 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
23 rabn0 3991 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
24 ssrab2 3720 . . . . . . . . . 10 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
2524biantrur 526 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2623, 25bitr3i 266 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2722, 26bitr3i 266 . . . . . . 7 (¬ ∀𝑥𝐴 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
28 fri 5105 . . . . . . 7 ((({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
2921, 27, 28syl2anb 495 . . . . . 6 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
30 eqid 2651 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
3130bnj23 30912 . . . . . . 7 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧 → ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
32 df-ral 2946 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
3332sbcbii 3524 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ [𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
34 sbcal 3518 . . . . . . . . . 10 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
35 sbcimg 3510 . . . . . . . . . . . . 13 (𝑧 ∈ V → ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))))
362, 35ax-mp 5 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
37 nfv 1883 . . . . . . . . . . . . . . 15 𝑥 𝑦𝐴
3837sbcgf 3534 . . . . . . . . . . . . . 14 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑦𝐴𝑦𝐴))
392, 38ax-mp 5 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑦𝐴𝑦𝐴)
40 sbcimg 3510 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑)))
412, 40ax-mp 5 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑))
42 sbcbr2g 4743 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥))
432, 42ax-mp 5 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥)
44 csbvarg 4036 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ V → 𝑧 / 𝑥𝑥 = 𝑧)
452, 44ax-mp 5 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥𝑥 = 𝑧
4645breq2i 4693 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧 / 𝑥𝑥𝑦𝑅𝑧)
4743, 46bitri 264 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧)
48 nfsbc1v 3488 . . . . . . . . . . . . . . . . 17 𝑥[𝑦 / 𝑥]𝜑
4948sbcgf 3534 . . . . . . . . . . . . . . . 16 (𝑧 ∈ V → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
502, 49ax-mp 5 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
5147, 50imbi12i 339 . . . . . . . . . . . . . 14 (([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5241, 51bitri 264 . . . . . . . . . . . . 13 ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5339, 52imbi12i 339 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5436, 53bitri 264 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5554albii 1787 . . . . . . . . . 10 (∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5634, 55bitri 264 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5733, 56bitri 264 . . . . . . . 8 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
58 bnj110.2 . . . . . . . . 9 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
5958sbcbii 3524 . . . . . . . 8 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
60 df-ral 2946 . . . . . . . 8 (∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
6157, 59, 603bitr4i 292 . . . . . . 7 ([𝑧 / 𝑥]𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
6231, 61sylibr 224 . . . . . 6 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧[𝑧 / 𝑥]𝜓)
6329, 62bnj31 30913 . . . . 5 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓)
64 nfv 1883 . . . . . . . 8 𝑧(𝜓𝜑)
65 nfsbc1v 3488 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜓
66 nfsbc1v 3488 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
6765, 66nfim 1865 . . . . . . . 8 𝑥([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)
68 sbceq1a 3479 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜓[𝑧 / 𝑥]𝜓))
69 sbceq1a 3479 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
7068, 69imbi12d 333 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝜑) ↔ ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
7164, 67, 70cbvral 3197 . . . . . . 7 (∀𝑥𝐴 (𝜓𝜑) ↔ ∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
72 elrabi 3391 . . . . . . . . 9 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → 𝑧𝐴)
7372imim1i 63 . . . . . . . 8 ((𝑧𝐴 → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)) → (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
7473ralimi2 2978 . . . . . . 7 (∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
7571, 74sylbi 207 . . . . . 6 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
76 rexim 3037 . . . . . 6 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7775, 76syl 17 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7863, 77mpan9 485 . . . 4 (((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
7978an32s 863 . . 3 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
8018, 79mto 188 . 2 ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑)
81 iman 439 . 2 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑) ↔ ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑))
8280, 81mpbir 221 1 ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  [wsbc 3468  csb 3566  wss 3607  c0 3948   class class class wbr 4685   Fr wfr 5099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-fr 5102
This theorem is referenced by:  bnj157  31055  bnj580  31109  bnj1052  31169  bnj1030  31181  bnj1133  31183
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