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Theorem undif3VD 43154
Description: The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 4250. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 4250 is undif3VD 43154 without virtual deductions and was automatically derived from undif3VD 43154.
1:: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴 𝑥 ∈ (𝐵𝐶)))
2:: (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥 𝐶))
3:2: ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4:1,3: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
5:: (   𝑥𝐴   ▶   𝑥𝐴   )
6:5: (   𝑥𝐴   ▶   (𝑥𝐴𝑥𝐵)   )
7:5: (   𝑥𝐴   ▶   𝑥𝐶𝑥𝐴)   )
8:6,7: (   𝑥𝐴   ▶   ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶𝑥𝐴))   )
9:8: (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ ( ¬ 𝑥𝐶𝑥𝐴)))
10:: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   )
11:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐵   )
12:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   ¬ 𝑥𝐶    )
13:11: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 𝑥𝐵)   )
14:12: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥 𝐶𝑥𝐴)   )
15:13,14: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   ((𝑥 𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴))   )
16:15: ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴 𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
17:9,16: ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
18:: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   )
19:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐴   )
20:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   ¬ 𝑥𝐶    )
21:18: (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
22:21: ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
23:: (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴 𝑥𝐴)   )
24:23: (   (𝑥𝐴𝑥𝐴)   ▶   𝑥𝐴   )
25:24: (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
26:25: ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ ( 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
27:10: (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
28:27: ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
29:: (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐵 𝑥𝐴)   )
30:29: (   (𝑥𝐵𝑥𝐴)   ▶   𝑥𝐴   )
31:30: (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐴 (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
32:31: ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ ( 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
33:22,26: (((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
34:28,32: (((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
35:33,34: ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥 𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
36:: ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥 𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
37:36,35: (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶 𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
38:17,37: ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
39:: (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥 𝐴))
40:39: 𝑥 ∈ (𝐶𝐴) ↔ ¬ (𝑥𝐶 ¬ 𝑥𝐴))
41:: (¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴) ↔ (¬ 𝑥 𝐶𝑥𝐴))
42:40,41: 𝑥 ∈ (𝐶𝐴) ↔ (¬ 𝑥𝐶𝑥 𝐴))
43:: (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵 ))
44:43,42: ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴) ) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
45:: (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ( 𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
46:45,44: (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ( (𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
47:4,38: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴 𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
48:46,47: (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴 𝐵) ∖ (𝐶𝐴)))
49:48: 𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ((𝐴𝐵) ∖ (𝐶𝐴)))
qed:49: (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶 𝐴))
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
undif3VD (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))

Proof of Theorem undif3VD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 4108 . . . . . 6 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3920 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 911 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
41, 3bitri 274 . . . . 5 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
5 idn1 42846 . . . . . . . . . 10 (   𝑥𝐴   ▶   𝑥𝐴   )
6 orc 865 . . . . . . . . . 10 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
75, 6e1a 42899 . . . . . . . . 9 (   𝑥𝐴   ▶   (𝑥𝐴𝑥𝐵)   )
8 olc 866 . . . . . . . . . 10 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
95, 8e1a 42899 . . . . . . . . 9 (   𝑥𝐴   ▶   𝑥𝐶𝑥𝐴)   )
10 pm3.2 470 . . . . . . . . 9 ((𝑥𝐴𝑥𝐵) → ((¬ 𝑥𝐶𝑥𝐴) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴))))
117, 9, 10e11 42960 . . . . . . . 8 (   𝑥𝐴   ▶   ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴))   )
1211in1 42843 . . . . . . 7 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
13 idn1 42846 . . . . . . . . . . 11 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   )
14 simpl 483 . . . . . . . . . . 11 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → 𝑥𝐵)
1513, 14e1a 42899 . . . . . . . . . 10 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐵   )
16 olc 866 . . . . . . . . . 10 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
1715, 16e1a 42899 . . . . . . . . 9 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴𝑥𝐵)   )
18 simpr 485 . . . . . . . . . . 11 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ¬ 𝑥𝐶)
1913, 18e1a 42899 . . . . . . . . . 10 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶    ¬ 𝑥𝐶   )
20 orc 865 . . . . . . . . . 10 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
2119, 20e1a 42899 . . . . . . . . 9 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   𝑥𝐶𝑥𝐴)   )
2217, 21, 10e11 42960 . . . . . . . 8 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴))   )
2322in1 42843 . . . . . . 7 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
2412, 23jaoi 855 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
25 anddi 1009 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) ↔ (((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))))
2625bicomi 223 . . . . . . 7 ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
27 idn1 42846 . . . . . . . . . . 11 (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   )
28 simpl 483 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
2928orcd 871 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3027, 29e1a 42899 . . . . . . . . . 10 (   (𝑥𝐴 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
3130in1 42843 . . . . . . . . 9 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
32 idn1 42846 . . . . . . . . . . . 12 (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴𝑥𝐴)   )
33 simpl 483 . . . . . . . . . . . 12 ((𝑥𝐴𝑥𝐴) → 𝑥𝐴)
3432, 33e1a 42899 . . . . . . . . . . 11 (   (𝑥𝐴𝑥𝐴)   ▶   𝑥𝐴   )
35 orc 865 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3634, 35e1a 42899 . . . . . . . . . 10 (   (𝑥𝐴𝑥𝐴)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
3736in1 42843 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3831, 37jaoi 855 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
39 olc 866 . . . . . . . . . . 11 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4013, 39e1a 42899 . . . . . . . . . 10 (   (𝑥𝐵 ∧ ¬ 𝑥𝐶)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
4140in1 42843 . . . . . . . . 9 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
42 idn1 42846 . . . . . . . . . . . 12 (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐵𝑥𝐴)   )
43 simpr 485 . . . . . . . . . . . 12 ((𝑥𝐵𝑥𝐴) → 𝑥𝐴)
4442, 43e1a 42899 . . . . . . . . . . 11 (   (𝑥𝐵𝑥𝐴)   ▶   𝑥𝐴   )
4544, 35e1a 42899 . . . . . . . . . 10 (   (𝑥𝐵𝑥𝐴)   ▶   (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶))   )
4645in1 42843 . . . . . . . . 9 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4741, 46jaoi 855 . . . . . . . 8 (((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4838, 47jaoi 855 . . . . . . 7 ((((𝑥𝐴 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐴𝑥𝐴)) ∨ ((𝑥𝐵 ∧ ¬ 𝑥𝐶) ∨ (𝑥𝐵𝑥𝐴))) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4926, 48sylbir 234 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
5024, 49impbii 208 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
514, 50bitri 274 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
52 eldif 3920 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
53 elun 4108 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
54 eldif 3920 . . . . . . . 8 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
5554notbii 319 . . . . . . 7 𝑥 ∈ (𝐶𝐴) ↔ ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
56 pm4.53 984 . . . . . . 7 (¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
5755, 56bitri 274 . . . . . 6 𝑥 ∈ (𝐶𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
5853, 57anbi12i 627 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
5952, 58bitri 274 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
6051, 59bitr4i 277 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
6160ax-gen 1797 . 2 𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
62 dfcleq 2729 . . 3 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴))))
6362biimpri 227 . 2 (∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴))) → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴)))
6461, 63e0a 43044 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 845  wal 1539   = wceq 1541  wcel 2106  cdif 3907  cun 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-dif 3913  df-un 3915  df-vd1 42842
This theorem is referenced by: (None)
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