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Theorem muc0or 6252
 Description: The cardinal product of two cardinal numbers is zero iff one of the numbers is zero. Biconditional form of theorem XI.2.34 of [Rosser] p. 380. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
muc0or NC NC ·c 0c 0c 0c

Proof of Theorem muc0or
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . . . . 5 NC Nc
2 elncs 6119 . . . . 5 NC Nc
31, 2anbi12i 678 . . . 4 NC NC Nc Nc
4 eeanv 1913 . . . 4 Nc Nc Nc Nc
53, 4bitr4i 243 . . 3 NC NC Nc Nc
6 vex 2862 . . . . . . . 8
7 vex 2862 . . . . . . . 8
86, 7mucnc 6131 . . . . . . 7 Nc ·c Nc Nc
9 df0c2 6137 . . . . . . 7 0c Nc
108, 9eqeq12i 2366 . . . . . 6 Nc ·c Nc 0c Nc Nc
116, 7xpex 5115 . . . . . . . . 9
1211eqnc 6127 . . . . . . . 8 Nc Nc
13 en0 6042 . . . . . . . 8
1412, 13bitri 240 . . . . . . 7 Nc Nc
15 xpeq0 5046 . . . . . . . 8
16 nceq 6108 . . . . . . . . 9 Nc Nc
17 nceq 6108 . . . . . . . . 9 Nc Nc
1816, 17orim12i 502 . . . . . . . 8 Nc Nc Nc Nc
1915, 18sylbi 187 . . . . . . 7 Nc Nc Nc Nc
2014, 19sylbi 187 . . . . . 6 Nc Nc Nc Nc Nc Nc
2110, 20sylbi 187 . . . . 5 Nc ·c Nc 0c Nc Nc Nc Nc
22 oveq12 5532 . . . . . . 7 Nc Nc ·c Nc ·c Nc
2322eqeq1d 2361 . . . . . 6 Nc Nc ·c 0c Nc ·c Nc 0c
24 eqeq1 2359 . . . . . . . . 9 Nc 0c Nc 0c
2524adantr 451 . . . . . . . 8 Nc Nc 0c Nc 0c
269eqeq2i 2363 . . . . . . . 8 Nc 0c Nc Nc
2725, 26syl6bb 252 . . . . . . 7 Nc Nc 0c Nc Nc
28 eqeq1 2359 . . . . . . . . 9 Nc 0c Nc 0c
299eqeq2i 2363 . . . . . . . . 9 Nc 0c Nc Nc
3028, 29syl6bb 252 . . . . . . . 8 Nc 0c Nc Nc
3130adantl 452 . . . . . . 7 Nc Nc 0c Nc Nc
3227, 31orbi12d 690 . . . . . 6 Nc Nc 0c 0c Nc Nc Nc Nc
3323, 32imbi12d 311 . . . . 5 Nc Nc ·c 0c 0c 0c Nc ·c Nc 0c Nc Nc Nc Nc
3421, 33mpbiri 224 . . . 4 Nc Nc ·c 0c 0c 0c
3534exlimivv 1635 . . 3 Nc Nc ·c 0c 0c 0c
365, 35sylbi 187 . 2 NC NC ·c 0c 0c 0c
37 0cnc 6138 . . . . . . 7 0c NC
38 muccom 6134 . . . . . . 7 0c NC NC 0c ·c ·c 0c
3937, 38mpan 651 . . . . . 6 NC 0c ·c ·c 0c
40 muc0 6142 . . . . . 6 NC ·c 0c 0c
4139, 40eqtrd 2385 . . . . 5 NC 0c ·c 0c
42 oveq1 5530 . . . . . 6 0c ·c 0c ·c
4342eqeq1d 2361 . . . . 5 0c ·c 0c 0c ·c 0c
4441, 43syl5ibrcom 213 . . . 4 NC 0c ·c 0c
4544adantl 452 . . 3 NC NC 0c ·c 0c
46 muc0 6142 . . . . 5 NC ·c 0c 0c
47 oveq2 5531 . . . . . 6 0c ·c ·c 0c
4847eqeq1d 2361 . . . . 5 0c ·c 0c ·c 0c 0c
4946, 48syl5ibrcom 213 . . . 4 NC 0c ·c 0c
5049adantr 451 . . 3 NC NC 0c ·c 0c
5145, 50jaod 369 . 2 NC NC 0c 0c ·c 0c
5236, 51impbid 183 1 NC NC ·c 0c 0c 0c
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wo 357   wa 358  wex 1541   wceq 1642   wcel 1710  c0 3550  0cc0c 4374   class class class wbr 4639   cxp 4770  (class class class)co 5525   cen 6028   NC cncs 6088   Nc cnc 6091   ·c cmuc 6092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-cross 5764  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-muc 6102 This theorem is referenced by: (None)
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