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

Step	Hyp	Ref	Expression
1		ovex 5551	. . 3 ⊢ ( Nc V ·c Nc V) ∈ V
2		nulnnc 6118	. . . . 5 ⊢ ¬ ∅ ∈ NC
3		vvex 4109	. . . . . . . 8 ⊢ V ∈ V
4	3	ncelncsi 6121	. . . . . . 7 ⊢ Nc V ∈ NC
5		muccl 6132	. . . . . . 7 ⊢ (( Nc V ∈ NC ∧ Nc V ∈ NC ) → ( Nc V ·c Nc V) ∈ NC )
6	4, 4, 5	mp2an 653	. . . . . 6 ⊢ ( Nc V ·c Nc V) ∈ NC
7		eleq1 2413	. . . . . 6 ⊢ (( Nc V ·c Nc V) = ∅ → (( Nc V ·c Nc V) ∈ NC ↔ ∅ ∈ NC ))
8	6, 7	mpbii 202	. . . . 5 ⊢ (( Nc V ·c Nc V) = ∅ → ∅ ∈ NC )
9	2, 8	mto 167	. . . 4 ⊢ ¬ ( Nc V ·c Nc V) = ∅
10		df-ne 2518	. . . 4 ⊢ (( Nc V ·c Nc V) ≠ ∅ ↔ ¬ ( Nc V ·c Nc V) = ∅)
11	9, 10	mpbir 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)
13	1, 11, 12	mp2an 653	. 2 ⊢ ( Nc V ·c Nc V) ≤c Nc V
14		vn0 3557	. . . . . 6 ⊢ V ≠ ∅
15		el0c 4421	. . . . . 6 ⊢ (V ∈ 0c ↔ V = ∅)
16	14, 15	nemtbir 2604	. . . . 5 ⊢ ¬ V ∈ 0c
17	3	ncid 6123	. . . . . 6 ⊢ V ∈ Nc V
18		eleq2 2414	. . . . . 6 ⊢ ( Nc V = 0c → (V ∈ Nc V ↔ V ∈ 0c))
19	17, 18	mpbii 202	. . . . 5 ⊢ ( Nc V = 0c → V ∈ 0c)
20	16, 19	mto 167	. . . 4 ⊢ ¬ Nc V = 0c
21		df-ne 2518	. . . 4 ⊢ ( Nc V ≠ 0c ↔ ¬ Nc V = 0c)
22	20, 21	mpbir 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))
24	4, 4, 22, 23	mp3an 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))
26	6, 4, 25	mp2an 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)
27	13, 24, 26	mp2an 653	1 ⊢ ( Nc V ·c Nc V) = Nc V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859  ∅c0 3550  0_cc0c 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-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-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)