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Theorem dflec2 6210
 Description: Cardinal less than or equal in terms of cardinal addition. Theorem XI.2.22 of [Rosser] p. 377. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
dflec2 ((M NC N NC ) → (Mc Np NC N = (M +c p)))
Distinct variable groups:   M,p   N,p

Proof of Theorem dflec2
Dummy variables a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brlecg 6112 . . 3 ((M NC N NC ) → (Mc Na M b N a b))
2 ncseqnc 6128 . . . . . . 7 (M NC → (M = Nc aa M))
3 ncseqnc 6128 . . . . . . 7 (N NC → (N = Nc bb N))
42, 3bi2anan9 843 . . . . . 6 ((M NC N NC ) → ((M = Nc a N = Nc b) ↔ (a M b N)))
54biimpar 471 . . . . 5 (((M NC N NC ) (a M b N)) → (M = Nc a N = Nc b))
6 vex 2862 . . . . . . . . 9 b V
7 vex 2862 . . . . . . . . 9 a V
86, 7difex 4107 . . . . . . . 8 (b a) V
98ncelncsi 6121 . . . . . . 7 Nc (b a) NC
10 disjdif 3622 . . . . . . . . 9 (a ∩ (b a)) =
117, 8ncdisjun 6136 . . . . . . . . 9 ((a ∩ (b a)) = Nc (a ∪ (b a)) = ( Nc a +c Nc (b a)))
1210, 11ax-mp 8 . . . . . . . 8 Nc (a ∪ (b a)) = ( Nc a +c Nc (b a))
13 undif2 3626 . . . . . . . . . 10 (a ∪ (b a)) = (ab)
14 ssequn1 3433 . . . . . . . . . . 11 (a b ↔ (ab) = b)
1514biimpi 186 . . . . . . . . . 10 (a b → (ab) = b)
1613, 15syl5eq 2397 . . . . . . . . 9 (a b → (a ∪ (b a)) = b)
1716nceqd 6110 . . . . . . . 8 (a bNc (a ∪ (b a)) = Nc b)
1812, 17syl5reqr 2400 . . . . . . 7 (a bNc b = ( Nc a +c Nc (b a)))
19 addceq2 4384 . . . . . . . . 9 (p = Nc (b a) → ( Nc a +c p) = ( Nc a +c Nc (b a)))
2019eqeq2d 2364 . . . . . . . 8 (p = Nc (b a) → ( Nc b = ( Nc a +c p) ↔ Nc b = ( Nc a +c Nc (b a))))
2120rspcev 2955 . . . . . . 7 (( Nc (b a) NC Nc b = ( Nc a +c Nc (b a))) → p NC Nc b = ( Nc a +c p))
229, 18, 21sylancr 644 . . . . . 6 (a bp NC Nc b = ( Nc a +c p))
23 id 19 . . . . . . . . 9 (N = Nc bN = Nc b)
24 addceq1 4383 . . . . . . . . 9 (M = Nc a → (M +c p) = ( Nc a +c p))
2523, 24eqeqan12d 2368 . . . . . . . 8 ((N = Nc b M = Nc a) → (N = (M +c p) ↔ Nc b = ( Nc a +c p)))
2625rexbidv 2635 . . . . . . 7 ((N = Nc b M = Nc a) → (p NC N = (M +c p) ↔ p NC Nc b = ( Nc a +c p)))
2726ancoms 439 . . . . . 6 ((M = Nc a N = Nc b) → (p NC N = (M +c p) ↔ p NC Nc b = ( Nc a +c p)))
2822, 27syl5ibr 212 . . . . 5 ((M = Nc a N = Nc b) → (a bp NC N = (M +c p)))
295, 28syl 15 . . . 4 (((M NC N NC ) (a M b N)) → (a bp NC N = (M +c p)))
3029rexlimdvva 2745 . . 3 ((M NC N NC ) → (a M b N a bp NC N = (M +c p)))
311, 30sylbid 206 . 2 ((M NC N NC ) → (Mc Np NC N = (M +c p)))
32 addlecncs 6209 . . . . 5 ((M NC p NC ) → Mc (M +c p))
33 breq2 4643 . . . . 5 (N = (M +c p) → (Mc NMc (M +c p)))
3432, 33syl5ibrcom 213 . . . 4 ((M NC p NC ) → (N = (M +c p) → Mc N))
3534adantlr 695 . . 3 (((M NC N NC ) p NC ) → (N = (M +c p) → Mc N))
3635rexlimdva 2738 . 2 ((M NC N NC ) → (p NC N = (M +c p) → Mc N))
3731, 36impbid 183 1 ((M NC N NC ) → (Mc Np NC N = (M +c p)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089   Nc cnc 6091 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-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-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-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  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 This theorem is referenced by:  lectr  6211  leaddc1  6214  nc0suc  6217  leconnnc  6218  tlecg  6230  letc  6231  nclenn  6249  lemuc1  6253  lecadd2  6266  ncslesuc  6267  nchoicelem14  6302  nchoicelem17  6305
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