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Theorem uneqdifeqim 3302
Description: Two ways that and can "partition"  C (when and don't overlap and is a part of  C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
uneqdifeqim  C_  C  i^i  (/)  u.  C  C  \

Proof of Theorem uneqdifeqim
StepHypRef Expression
1 uncom 3081 . . . 4  u.  u.
2 eqtr 2054 . . . . . 6  u.  u.  u.  C  u.  C
32eqcomd 2042 . . . . 5  u.  u.  u.  C  C  u.
4 difeq1 3049 . . . . . 6  C  u.  C  \  u.  \
5 difun2 3296 . . . . . 6  u. 
\  \
6 eqtr 2054 . . . . . . 7  C  \  u.  \  u.  \  \  C  \  \
7 incom 3123 . . . . . . . . . 10  i^i  i^i
87eqeq1i 2044 . . . . . . . . 9  i^i  (/)  i^i  (/)
9 disj3 3266 . . . . . . . . 9  i^i  (/)  \
108, 9bitri 173 . . . . . . . 8  i^i  (/)  \
11 eqtr 2054 . . . . . . . . . 10  C  \  \  \  C  \
1211expcom 109 . . . . . . . . 9  \  C  \  \  C  \
1312eqcoms 2040 . . . . . . . 8  \  C  \  \  C  \
1410, 13sylbi 114 . . . . . . 7  i^i  (/)  C  \  \  C  \
156, 14syl5com 26 . . . . . 6  C  \  u.  \  u.  \  \  i^i  (/)  C  \
164, 5, 15sylancl 392 . . . . 5  C  u.  i^i  (/)  C  \
173, 16syl 14 . . . 4  u.  u.  u.  C  i^i  (/)  C  \
181, 17mpan 400 . . 3  u.  C  i^i  (/)  C  \
1918com12 27 . 2  i^i  (/)  u.  C  C  \
2019adantl 262 1  C_  C  i^i  (/)  u.  C  C  \
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242    \ cdif 2908    u. cun 2909    i^i cin 2910    C_ wss 2911   (/)c0 3218
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-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by: (None)
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