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Theorem undif3ss 3368
 Description: A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
undif3ss

Proof of Theorem undif3ss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elun 3248 . . . 4
2 eldif 3111 . . . . 5
32orbi2i 752 . . . 4
4 orc 702 . . . . . . 7
5 olc 701 . . . . . . 7
64, 5jca 304 . . . . . 6
7 olc 701 . . . . . . 7
8 orc 702 . . . . . . 7
97, 8anim12i 336 . . . . . 6
106, 9jaoi 706 . . . . 5
11 simpl 108 . . . . . . 7
1211orcd 723 . . . . . 6
13 olc 701 . . . . . 6
14 orc 702 . . . . . . 7
1514adantr 274 . . . . . 6
1614adantl 275 . . . . . 6
1712, 13, 15, 16ccase 949 . . . . 5
1810, 17impbii 125 . . . 4
191, 3, 183bitri 205 . . 3
20 elun 3248 . . . . . 6
2120biimpri 132 . . . . 5
22 pm4.53r 741 . . . . . 6
23 eldif 3111 . . . . . 6
2422, 23sylnibr 667 . . . . 5
2521, 24anim12i 336 . . . 4
26 eldif 3111 . . . 4
2725, 26sylibr 133 . . 3
2819, 27sylbi 120 . 2
2928ssriv 3132 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 103   wo 698   wcel 2128   cdif 3099   cun 3100   wss 3102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115 This theorem is referenced by: (None)
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