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Theorem undif3ss 3192
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  u.  \  C  C_  u.  \  C  \

Proof of Theorem undif3ss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elun 3078 . . . 4  u.  \  C  \  C
2 eldif 2921 . . . . 5  \  C  C
32orbi2i 678 . . . 4  \  C  C
4 orc 632 . . . . . . 7
5 olc 631 . . . . . . 7  C
64, 5jca 290 . . . . . 6  C
7 olc 631 . . . . . . 7
8 orc 632 . . . . . . 7  C  C
97, 8anim12i 321 . . . . . 6  C  C
106, 9jaoi 635 . . . . 5  C  C
11 simpl 102 . . . . . . 7  C
1211orcd 651 . . . . . 6  C  C
13 olc 631 . . . . . 6  C  C
14 orc 632 . . . . . . 7  C
1514adantr 261 . . . . . 6  C
1614adantl 262 . . . . . 6  C
1712, 13, 15, 16ccase 870 . . . . 5  C  C
1810, 17impbii 117 . . . 4  C  C
191, 3, 183bitri 195 . . 3  u.  \  C  C
20 elun 3078 . . . . . 6  u.
2120biimpri 124 . . . . 5  u.
22 pm4.53r 803 . . . . . 6  C  C
23 eldif 2921 . . . . . 6  C  \  C
2422, 23sylnibr 601 . . . . 5  C  C  \
2521, 24anim12i 321 . . . 4  C  u.  C  \
26 eldif 2921 . . . 4  u.  \  C  \  u.  C  \
2725, 26sylibr 137 . . 3  C  u.  \  C  \
2819, 27sylbi 114 . 2  u.  \  C  u.  \  C  \
2928ssriv 2943 1  u.  \  C  C_  u.  \  C  \
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wo 628   wcel 1390    \ cdif 2908    u. cun 2909    C_ wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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