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Theorem ncvsq 6256
Description: The product of the cardinality of V squared is just the cardinality of V. Theorem XI.2.37 of [Rosser] p. 381. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
ncvsq ( Nc V ·c Nc V) = Nc V

Proof of Theorem ncvsq
StepHypRef Expression
1 ovex 5551 . . 3 ( Nc V ·c Nc V) V
2 nulnnc 6118 . . . . 5 ¬ NC
3 vvex 4109 . . . . . . . 8 V V
43ncelncsi 6121 . . . . . . 7 Nc V NC
5 muccl 6132 . . . . . . 7 (( Nc V NC Nc V NC ) → ( Nc V ·c Nc V) NC )
64, 4, 5mp2an 653 . . . . . 6 ( Nc V ·c Nc V) NC
7 eleq1 2413 . . . . . 6 (( Nc V ·c Nc V) = → (( Nc V ·c Nc V) NC NC ))
86, 7mpbii 202 . . . . 5 (( Nc V ·c Nc V) = NC )
92, 8mto 167 . . . 4 ¬ ( Nc V ·c Nc V) =
10 df-ne 2518 . . . 4 (( Nc V ·c Nc V) ≠ ↔ ¬ ( Nc V ·c Nc V) = )
119, 10mpbir 200 . . 3 ( Nc V ·c Nc V) ≠
12 lecncvg 6199 . . 3 ((( Nc V ·c Nc V) V ( Nc V ·c Nc V) ≠ ) → ( Nc V ·c Nc V) ≤c Nc V)
131, 11, 12mp2an 653 . 2 ( Nc V ·c Nc V) ≤c Nc V
14 vn0 3557 . . . . . 6 V ≠
15 el0c 4421 . . . . . 6 (V 0c ↔ V = )
1614, 15nemtbir 2604 . . . . 5 ¬ V 0c
173ncid 6123 . . . . . 6 V Nc V
18 eleq2 2414 . . . . . 6 ( Nc V = 0c → (V Nc V ↔ V 0c))
1917, 18mpbii 202 . . . . 5 ( Nc V = 0c → V 0c)
2016, 19mto 167 . . . 4 ¬ Nc V = 0c
21 df-ne 2518 . . . 4 ( Nc V ≠ 0c ↔ ¬ Nc V = 0c)
2220, 21mpbir 200 . . 3 Nc V ≠ 0c
23 ncslemuc 6255 . . 3 (( Nc V NC Nc V NC Nc V ≠ 0c) → Nc V ≤c ( Nc V ·c Nc V))
244, 4, 22, 23mp3an 1277 . 2 Nc V ≤c ( Nc V ·c Nc V)
25 sbth 6206 . . 3 ((( Nc V ·c Nc V) NC Nc V NC ) → ((( Nc V ·c Nc V) ≤c Nc V Nc V ≤c ( Nc V ·c Nc V)) → ( Nc V ·c Nc V) = Nc V))
266, 4, 25mp2an 653 . 2 ((( Nc V ·c Nc V) ≤c Nc V Nc V ≤c ( Nc V ·c Nc V)) → ( Nc V ·c Nc V) = Nc V)
2713, 24, 26mp2an 653 1 ( Nc V ·c Nc V) = Nc V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wa 358   = wceq 1642   wcel 1710  wne 2516  Vcvv 2859  c0 3550  0cc0c 4374   class class class wbr 4639  (class class class)co 5525   NC cncs 6088  c clec 6089   Nc cnc 6091   ·c cmuc 6092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-fix 5740  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-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101  df-muc 6102
This theorem is referenced by: (None)
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