Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj110 Structured version   Visualization version   GIF version

Theorem bnj110 31463
 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 3201 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
2 vex 3417 . . . . . . . 8 𝑧 ∈ V
3 sbcng 3703 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑))
42, 3ax-mp 5 . . . . . . 7 ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑)
54bicomi 216 . . . . . 6 [𝑧 / 𝑥]𝜑[𝑧 / 𝑥] ¬ 𝜑)
65ralbii 3189 . . . . 5 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
71, 6bitr3i 269 . . . 4 (¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥] ¬ 𝜑)
8 df-rab 3126 . . . . . . 7 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
98eleq2i 2898 . . . . . 6 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
10 df-sbc 3663 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)})
11 sbcan 3705 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑))
12 sbcel1v 3721 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
1312anbi1i 617 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴[𝑧 / 𝑥] ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1411, 13bitri 267 . . . . . . 7 ([𝑧 / 𝑥](𝑥𝐴 ∧ ¬ 𝜑) ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1510, 14bitr3i 269 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
169, 15bitri 267 . . . . 5 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ↔ (𝑧𝐴[𝑧 / 𝑥] ¬ 𝜑))
1716simprbi 492 . . . 4 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → [𝑧 / 𝑥] ¬ 𝜑)
187, 17mprgbir 3136 . . 3 ¬ ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑
19 bnj110.1 . . . . . . . . 9 𝐴 ∈ V
2019rabex 5037 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} ∈ V
2120biantrur 526 . . . . . . 7 (𝑅 Fr 𝐴 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴))
22 rexnal 3203 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
23 rabn0 4187 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥𝐴 ¬ 𝜑)
24 ssrab2 3912 . . . . . . . . . 10 {𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
2524biantrur 526 . . . . . . . . 9 ({𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2623, 25bitr3i 269 . . . . . . . 8 (∃𝑥𝐴 ¬ 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
2722, 26bitr3i 269 . . . . . . 7 (¬ ∀𝑥𝐴 𝜑 ↔ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅))
28 fri 5304 . . . . . . 7 ((({𝑥𝐴 ∣ ¬ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥𝐴 ∣ ¬ 𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
2921, 27, 28syl2anb 591 . . . . . 6 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧)
30 eqid 2825 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐴 ∣ ¬ 𝜑}
3130bnj23 31322 . . . . . . 7 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧 → ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
32 df-ral 3122 . . . . . . . . . 10 (∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
3332sbcbii 3718 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ [𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
34 sbcal 3712 . . . . . . . . . 10 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)))
35 sbcimg 3704 . . . . . . . . . . . . 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 2013 . . . . . . . . . . . . . . 15 𝑥 𝑦𝐴
3837sbcgf 3726 . . . . . . . . . . . . . 14 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑦𝐴𝑦𝐴))
392, 38ax-mp 5 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑦𝐴𝑦𝐴)
40 sbcimg 3704 . . . . . . . . . . . . . . 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 4931 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥))
432, 42ax-mp 5 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧 / 𝑥𝑥)
442csbvargi 4228 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥𝑥 = 𝑧
4544breq2i 4881 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧 / 𝑥𝑥𝑦𝑅𝑧)
4643, 45bitri 267 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑅𝑥𝑦𝑅𝑧)
47 nfsbc1v 3682 . . . . . . . . . . . . . . . . 17 𝑥[𝑦 / 𝑥]𝜑
4847sbcgf 3726 . . . . . . . . . . . . . . . 16 (𝑧 ∈ V → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
492, 48ax-mp 5 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
5046, 49imbi12i 342 . . . . . . . . . . . . . 14 (([𝑧 / 𝑥]𝑦𝑅𝑥[𝑧 / 𝑥][𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5141, 50bitri 267 . . . . . . . . . . . . 13 ([𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
5239, 51imbi12i 342 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑦𝐴[𝑧 / 𝑥](𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5336, 52bitri 267 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ (𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5453albii 1918 . . . . . . . . . 10 (∀𝑦[𝑧 / 𝑥](𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5534, 54bitri 267 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦(𝑦𝐴 → (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑)) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
5633, 55bitri 267 . . . . . . . 8 ([𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
57 bnj110.2 . . . . . . . . 9 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
5857sbcbii 3718 . . . . . . . 8 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
59 df-ral 3122 . . . . . . . 8 (∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑)))
6056, 58, 593bitr4i 295 . . . . . . 7 ([𝑧 / 𝑥]𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑧[𝑦 / 𝑥]𝜑))
6131, 60sylibr 226 . . . . . 6 (∀𝑤 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ¬ 𝑤𝑅𝑧[𝑧 / 𝑥]𝜓)
6229, 61bnj31 31323 . . . . 5 ((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓)
63 nfv 2013 . . . . . . . 8 𝑧(𝜓𝜑)
64 nfsbc1v 3682 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜓
65 nfsbc1v 3682 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
6664, 65nfim 1999 . . . . . . . 8 𝑥([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)
67 sbceq1a 3673 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜓[𝑧 / 𝑥]𝜓))
68 sbceq1a 3673 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
6967, 68imbi12d 336 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝜑) ↔ ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
7063, 66, 69cbvral 3379 . . . . . . 7 (∀𝑥𝐴 (𝜓𝜑) ↔ ∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
71 elrabi 3580 . . . . . . . . 9 (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → 𝑧𝐴)
7271imim1i 63 . . . . . . . 8 ((𝑧𝐴 → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)) → (𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} → ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑)))
7372ralimi2 3158 . . . . . . 7 (∀𝑧𝐴 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
7470, 73sylbi 209 . . . . . 6 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑))
75 rexim 3216 . . . . . 6 (∀𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑} ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7674, 75syl 17 . . . . 5 (∀𝑥𝐴 (𝜓𝜑) → (∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜓 → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑))
7762, 76mpan9 502 . . . 4 (((𝑅 Fr 𝐴 ∧ ¬ ∀𝑥𝐴 𝜑) ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
7877an32s 642 . . 3 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑) → ∃𝑧 ∈ {𝑥𝐴 ∣ ¬ 𝜑}[𝑧 / 𝑥]𝜑)
7918, 78mto 189 . 2 ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑)
80 iman 392 . 2 (((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑) ↔ ¬ ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) ∧ ¬ ∀𝑥𝐴 𝜑))
8179, 80mpbir 223 1 ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654   ∈ wcel 2164  {cab 2811   ≠ wne 2999  ∀wral 3117  ∃wrex 3118  {crab 3121  Vcvv 3414  [wsbc 3662  ⦋csb 3757   ⊆ wss 3798  ∅c0 4144   class class class wbr 4873   Fr wfr 5298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-fr 5301 This theorem is referenced by:  bnj157  31464  bnj580  31518  bnj1052  31578  bnj1030  31590  bnj1133  31592
 Copyright terms: Public domain W3C validator