MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omopth2 Structured version   Unicode version

Theorem omopth2 7251
Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omopth2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )

Proof of Theorem omopth2
StepHypRef Expression
1 simpl2l 1049 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  e.  On )
2 eloni 4897 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 16 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  B )
4 simpl3l 1051 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  e.  On )
5 eloni 4897 . . . . . . 7  |-  ( D  e.  On  ->  Ord  D )
64, 5syl 16 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  D )
7 ordtri3or 4919 . . . . . 6  |-  ( ( Ord  B  /\  Ord  D )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
83, 6, 7syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
9 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E ) )
10 simpl1l 1047 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  e.  On )
11 omcl 7204 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
1210, 4, 11syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( A  .o  D )  e.  On )
13 simpl3r 1052 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  A
)
14 onelon 4912 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
1510, 13, 14syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  On )
16 oacl 7203 . . . . . . . . . . 11  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( ( A  .o  D )  +o  E
)  e.  On )
1712, 15, 16syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  D )  +o  E )  e.  On )
18 eloni 4897 . . . . . . . . . 10  |-  ( ( ( A  .o  D
)  +o  E )  e.  On  ->  Ord  ( ( A  .o  D )  +o  E
) )
19 ordirr 4905 . . . . . . . . . 10  |-  ( Ord  ( ( A  .o  D )  +o  E
)  ->  -.  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  D )  +o  E ) )
2017, 18, 193syl 20 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  D
)  +o  E ) )
219, 20eqneltrd 2566 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) )
22 orc 385 . . . . . . . . 9  |-  ( B  e.  D  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
23 omeulem2 7250 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
2423adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E )
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2522, 24syl5 32 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2621, 25mtod 177 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  B  e.  D )
2726pm2.21d 106 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  B  =  D ) )
28 idd 24 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  ->  B  =  D ) )
2920, 9neleqtrrd 2570 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) )
30 orc 385 . . . . . . . . 9  |-  ( D  e.  B  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
31 simpl1r 1048 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  =/=  (/) )
32 simpl2r 1050 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  e.  A
)
33 omeulem2 7250 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( D  e.  On  /\  E  e.  A )  /\  ( B  e.  On  /\  C  e.  A ) )  -> 
( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C ) )  -> 
( ( A  .o  D )  +o  E
)  e.  ( ( A  .o  B )  +o  C ) ) )
3410, 31, 4, 13, 1, 32, 33syl222anc 1244 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C )
)  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3530, 34syl5 32 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3629, 35mtod 177 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  D  e.  B )
3736pm2.21d 106 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  B  =  D ) )
3827, 28, 373jaod 1292 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  B  =  D  \/  D  e.  B )  ->  B  =  D ) )
398, 38mpd 15 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  =  D )
40 onelon 4912 . . . . . . . 8  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
41 eloni 4897 . . . . . . . 8  |-  ( C  e.  On  ->  Ord  C )
4240, 41syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  A )  ->  Ord  C )
4310, 32, 42syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  C )
44 eloni 4897 . . . . . . . 8  |-  ( E  e.  On  ->  Ord  E )
4514, 44syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  E  e.  A )  ->  Ord  E )
4610, 13, 45syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  E )
47 ordtri3or 4919 . . . . . 6  |-  ( ( Ord  C  /\  Ord  E )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
4843, 46, 47syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
49 olc 384 . . . . . . . . . 10  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
5049, 24syl5 32 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  =  D  /\  C  e.  E )  ->  (
( A  .o  B
)  +o  C )  e.  ( ( A  .o  D )  +o  E ) ) )
5139, 50mpand 675 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
5221, 51mtod 177 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  C  e.  E )
5352pm2.21d 106 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  C  =  E ) )
54 idd 24 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  =  E  ->  C  =  E ) )
5539eqcomd 2465 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  =  B )
56 olc 384 . . . . . . . . . 10  |-  ( ( D  =  B  /\  E  e.  C )  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
5756, 34syl5 32 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  =  B  /\  E  e.  C )  ->  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  B )  +o  C ) ) )
5855, 57mpand 675 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
5929, 58mtod 177 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  E  e.  C )
6059pm2.21d 106 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  C  =  E ) )
6153, 54, 603jaod 1292 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( C  e.  E  \/  C  =  E  \/  E  e.  C )  ->  C  =  E ) )
6248, 61mpd 15 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  =  E )
6339, 62jca 532 . . 3  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  /\  C  =  E ) )
6463ex 434 . 2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  ->  ( B  =  D  /\  C  =  E ) ) )
65 oveq2 6304 . . 3  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
66 id 22 . . 3  |-  ( C  =  E  ->  C  =  E )
6765, 66oveqan12d 6315 . 2  |-  ( ( B  =  D  /\  C  =  E )  ->  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )
6864, 67impbid1 203 1  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   (/)c0 3793   Ord word 4886   Oncon0 4887  (class class class)co 6296    +o coa 7145    .o comu 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-oadd 7152  df-omul 7153
This theorem is referenced by:  omeu  7252  dfac12lem2  8541
  Copyright terms: Public domain W3C validator