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Theorem undif3VD 27348
Description: The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3336. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3336 is undif3VD 27348 without virtual deductions and was automatically derived from undif3VD 27348.
 1:: 2:: 3:2: 4:1,3: 5:: 6:5: 7:5: 8:6,7: 9:8: 10:: 11:10: 12:10: 13:11: 14:12: 15:13,14: 16:15: 17:9,16: 18:: 19:18: 20:18: 21:18: 22:21: 23:: 24:23: 25:24: 26:25: 27:10: 28:27: 29:: 30:29: 31:30: 32:31: 33:22,26: 34:28,32: 35:33,34: 36:: 37:36,35: 38:17,37: 39:: 40:39: 41:: 42:40,41: 43:: 44:43,42: 45:: 46:45,44: 47:4,38: 48:46,47: 49:48: qed:49:
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
undif3VD

Proof of Theorem undif3VD
StepHypRef Expression
1 elun 3226 . . . . . 6
2 eldif 3088 . . . . . . 7
32orbi2i 507 . . . . . 6
41, 3bitri 242 . . . . 5
5 idn1 27035 . . . . . . . . . 10
6 orc 376 . . . . . . . . . 10
75, 6e1_ 27089 . . . . . . . . 9
8 olc 375 . . . . . . . . . 10
95, 8e1_ 27089 . . . . . . . . 9
10 pm3.2 436 . . . . . . . . 9
117, 9, 10e11 27150 . . . . . . . 8
1211in1 27032 . . . . . . 7
13 idn1 27035 . . . . . . . . . . 11
14 simpl 445 . . . . . . . . . . 11
1513, 14e1_ 27089 . . . . . . . . . 10
16 olc 375 . . . . . . . . . 10
1715, 16e1_ 27089 . . . . . . . . 9
18 simpr 449 . . . . . . . . . . 11
1913, 18e1_ 27089 . . . . . . . . . 10
20 orc 376 . . . . . . . . . 10
2119, 20e1_ 27089 . . . . . . . . 9
2217, 21, 10e11 27150 . . . . . . . 8
2322in1 27032 . . . . . . 7
2412, 23jaoi 370 . . . . . 6
25 anddi 845 . . . . . . . 8
2625bicomi 195 . . . . . . 7
27 idn1 27035 . . . . . . . . . . 11
28 simpl 445 . . . . . . . . . . . 12
2928orcd 383 . . . . . . . . . . 11
3027, 29e1_ 27089 . . . . . . . . . 10
3130in1 27032 . . . . . . . . 9
32 idn1 27035 . . . . . . . . . . . 12
33 simpl 445 . . . . . . . . . . . 12
3432, 33e1_ 27089 . . . . . . . . . . 11
35 orc 376 . . . . . . . . . . 11
3634, 35e1_ 27089 . . . . . . . . . 10
3736in1 27032 . . . . . . . . 9
3831, 37jaoi 370 . . . . . . . 8
39 olc 375 . . . . . . . . . . 11
4013, 39e1_ 27089 . . . . . . . . . 10
4140in1 27032 . . . . . . . . 9
42 idn1 27035 . . . . . . . . . . . 12
43 simpr 449 . . . . . . . . . . . 12
4442, 43e1_ 27089 . . . . . . . . . . 11
4544, 35e1_ 27089 . . . . . . . . . 10
4645in1 27032 . . . . . . . . 9
4741, 46jaoi 370 . . . . . . . 8
4838, 47jaoi 370 . . . . . . 7
4926, 48sylbir 206 . . . . . 6
5024, 49impbii 182 . . . . 5
514, 50bitri 242 . . . 4
52 eldif 3088 . . . . 5
53 elun 3226 . . . . . 6
54 eldif 3088 . . . . . . . 8
5554notbii 289 . . . . . . 7
56 pm4.53 480 . . . . . . 7
5755, 56bitri 242 . . . . . 6
5853, 57anbi12i 681 . . . . 5
5952, 58bitri 242 . . . 4
6051, 59bitr4i 245 . . 3
6160ax-gen 1536 . 2
62 dfcleq 2247 . . 3
6362biimpri 199 . 2
6461, 63e0_ 27237 1
 Colors of variables: wff set class Syntax hints:   wn 5   wb 178   wo 359   wa 360  wal 1532   wceq 1619   wcel 1621   cdif 3075   cun 3076 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-un 3083  df-vd1 27031
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