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Theorem letc 6231
 Description: If a cardinal is less than or equal to a T-raising, then it is also a T-raising. Theorem 5.6 of [Specker] p. 973. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
letc ((M NC N NC Mc Tc N) → p NC M = Tc p)
Distinct variable group:   M,p
Allowed substitution hint:   N(p)

Proof of Theorem letc
Dummy variables a b c m n q x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tccl 6160 . . . 4 (N NCTc N NC )
2 dflec2 6210 . . . 4 ((M NC Tc N NC ) → (Mc Tc Nq NC Tc N = (M +c q)))
31, 2sylan2 460 . . 3 ((M NC N NC ) → (Mc Tc Nq NC Tc N = (M +c q)))
4 elncs 6119 . . . . . . . 8 (M NCa M = Nc a)
5 elncs 6119 . . . . . . . 8 (N NCb N = Nc b)
6 elncs 6119 . . . . . . . 8 (q NCc q = Nc c)
74, 5, 63anbi123i 1140 . . . . . . 7 ((M NC N NC q NC ) ↔ (a M = Nc a b N = Nc b c q = Nc c))
8 eeeanv 1914 . . . . . . 7 (abc(M = Nc a N = Nc b q = Nc c) ↔ (a M = Nc a b N = Nc b c q = Nc c))
97, 8bitr4i 243 . . . . . 6 ((M NC N NC q NC ) ↔ abc(M = Nc a N = Nc b q = Nc c))
10 eqcom 2355 . . . . . . . . . . 11 ( Nc 1b = ( Nc a +c Nc c) ↔ ( Nc a +c Nc c) = Nc 1b)
11 vex 2862 . . . . . . . . . . . . . 14 a V
1211ncelncsi 6121 . . . . . . . . . . . . 13 Nc a NC
13 vex 2862 . . . . . . . . . . . . . 14 c V
1413ncelncsi 6121 . . . . . . . . . . . . 13 Nc c NC
15 ncaddccl 6144 . . . . . . . . . . . . 13 (( Nc a NC Nc c NC ) → ( Nc a +c Nc c) NC )
1612, 14, 15mp2an 653 . . . . . . . . . . . 12 ( Nc a +c Nc c) NC
17 ncseqnc 6128 . . . . . . . . . . . 12 (( Nc a +c Nc c) NC → (( Nc a +c Nc c) = Nc 1b1b ( Nc a +c Nc c)))
1816, 17ax-mp 5 . . . . . . . . . . 11 (( Nc a +c Nc c) = Nc 1b1b ( Nc a +c Nc c))
1910, 18bitri 240 . . . . . . . . . 10 ( Nc 1b = ( Nc a +c Nc c) ↔ 1b ( Nc a +c Nc c))
20 eladdc 4398 . . . . . . . . . . 11 (1b ( Nc a +c Nc c) ↔ x Nc ay Nc c((xy) = 1b = (xy)))
21 vex 2862 . . . . . . . . . . . . . . 15 x V
22 vex 2862 . . . . . . . . . . . . . . 15 y V
2321, 22pw1equn 4331 . . . . . . . . . . . . . 14 (1b = (xy) ↔ nm(b = (nm) x = 1n y = 1m))
24 eleq1 2413 . . . . . . . . . . . . . . . . . . . 20 (x = 1n → (x Nc a1n Nc a))
25 eleq1 2413 . . . . . . . . . . . . . . . . . . . 20 (y = 1m → (y Nc c1m Nc c))
2624, 25bi2anan9 843 . . . . . . . . . . . . . . . . . . 19 ((x = 1n y = 1m) → ((x Nc a y Nc c) ↔ (1n Nc a 1m Nc c)))
27 ineq12 3452 . . . . . . . . . . . . . . . . . . . 20 ((x = 1n y = 1m) → (xy) = (1n1m))
2827eqeq1d 2361 . . . . . . . . . . . . . . . . . . 19 ((x = 1n y = 1m) → ((xy) = ↔ (1n1m) = ))
2926, 28anbi12d 691 . . . . . . . . . . . . . . . . . 18 ((x = 1n y = 1m) → (((x Nc a y Nc c) (xy) = ) ↔ ((1n Nc a 1m Nc c) (1n1m) = )))
30 ncseqnc 6128 . . . . . . . . . . . . . . . . . . . . 21 ( Nc a NC → ( Nc a = Nc 1n1n Nc a))
3112, 30ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ( Nc a = Nc 1n1n Nc a)
32 vex 2862 . . . . . . . . . . . . . . . . . . . . . 22 n V
3332ncelncsi 6121 . . . . . . . . . . . . . . . . . . . . 21 Nc n NC
34 tceq 6158 . . . . . . . . . . . . . . . . . . . . . . . 24 (p = Nc nTc p = Tc Nc n)
3532tcnc 6225 . . . . . . . . . . . . . . . . . . . . . . . 24 Tc Nc n = Nc 1n
3634, 35syl6eq 2401 . . . . . . . . . . . . . . . . . . . . . . 23 (p = Nc nTc p = Nc 1n)
3736eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . . 22 (p = Nc n → ( Nc a = Tc pNc a = Nc 1n))
3837rspcev 2955 . . . . . . . . . . . . . . . . . . . . 21 (( Nc n NC Nc a = Nc 1n) → p NC Nc a = Tc p)
3933, 38mpan 651 . . . . . . . . . . . . . . . . . . . 20 ( Nc a = Nc 1np NC Nc a = Tc p)
4031, 39sylbir 204 . . . . . . . . . . . . . . . . . . 19 (1n Nc ap NC Nc a = Tc p)
4140ad2antrr 706 . . . . . . . . . . . . . . . . . 18 (((1n Nc a 1m Nc c) (1n1m) = ) → p NC Nc a = Tc p)
4229, 41syl6bi 219 . . . . . . . . . . . . . . . . 17 ((x = 1n y = 1m) → (((x Nc a y Nc c) (xy) = ) → p NC Nc a = Tc p))
43423adant1 973 . . . . . . . . . . . . . . . 16 ((b = (nm) x = 1n y = 1m) → (((x Nc a y Nc c) (xy) = ) → p NC Nc a = Tc p))
4443exlimivv 1635 . . . . . . . . . . . . . . 15 (nm(b = (nm) x = 1n y = 1m) → (((x Nc a y Nc c) (xy) = ) → p NC Nc a = Tc p))
4544com12 27 . . . . . . . . . . . . . 14 (((x Nc a y Nc c) (xy) = ) → (nm(b = (nm) x = 1n y = 1m) → p NC Nc a = Tc p))
4623, 45syl5bi 208 . . . . . . . . . . . . 13 (((x Nc a y Nc c) (xy) = ) → (1b = (xy) → p NC Nc a = Tc p))
4746expimpd 586 . . . . . . . . . . . 12 ((x Nc a y Nc c) → (((xy) = 1b = (xy)) → p NC Nc a = Tc p))
4847rexlimivv 2743 . . . . . . . . . . 11 (x Nc ay Nc c((xy) = 1b = (xy)) → p NC Nc a = Tc p)
4920, 48sylbi 187 . . . . . . . . . 10 (1b ( Nc a +c Nc c) → p NC Nc a = Tc p)
5019, 49sylbi 187 . . . . . . . . 9 ( Nc 1b = ( Nc a +c Nc c) → p NC Nc a = Tc p)
51 tceq 6158 . . . . . . . . . . . . 13 (N = Nc bTc N = Tc Nc b)
52 vex 2862 . . . . . . . . . . . . . 14 b V
5352tcnc 6225 . . . . . . . . . . . . 13 Tc Nc b = Nc 1b
5451, 53syl6eq 2401 . . . . . . . . . . . 12 (N = Nc bTc N = Nc 1b)
55543ad2ant2 977 . . . . . . . . . . 11 ((M = Nc a N = Nc b q = Nc c) → Tc N = Nc 1b)
56 addceq12 4385 . . . . . . . . . . . 12 ((M = Nc a q = Nc c) → (M +c q) = ( Nc a +c Nc c))
57563adant2 974 . . . . . . . . . . 11 ((M = Nc a N = Nc b q = Nc c) → (M +c q) = ( Nc a +c Nc c))
5855, 57eqeq12d 2367 . . . . . . . . . 10 ((M = Nc a N = Nc b q = Nc c) → ( Tc N = (M +c q) ↔ Nc 1b = ( Nc a +c Nc c)))
59 eqeq1 2359 . . . . . . . . . . . 12 (M = Nc a → (M = Tc pNc a = Tc p))
6059rexbidv 2635 . . . . . . . . . . 11 (M = Nc a → (p NC M = Tc pp NC Nc a = Tc p))
61603ad2ant1 976 . . . . . . . . . 10 ((M = Nc a N = Nc b q = Nc c) → (p NC M = Tc pp NC Nc a = Tc p))
6258, 61imbi12d 311 . . . . . . . . 9 ((M = Nc a N = Nc b q = Nc c) → (( Tc N = (M +c q) → p NC M = Tc p) ↔ ( Nc 1b = ( Nc a +c Nc c) → p NC Nc a = Tc p)))
6350, 62mpbiri 224 . . . . . . . 8 ((M = Nc a N = Nc b q = Nc c) → ( Tc N = (M +c q) → p NC M = Tc p))
6463exlimiv 1634 . . . . . . 7 (c(M = Nc a N = Nc b q = Nc c) → ( Tc N = (M +c q) → p NC M = Tc p))
6564exlimivv 1635 . . . . . 6 (abc(M = Nc a N = Nc b q = Nc c) → ( Tc N = (M +c q) → p NC M = Tc p))
669, 65sylbi 187 . . . . 5 ((M NC N NC q NC ) → ( Tc N = (M +c q) → p NC M = Tc p))
67663expa 1151 . . . 4 (((M NC N NC ) q NC ) → ( Tc N = (M +c q) → p NC M = Tc p))
6867rexlimdva 2738 . . 3 ((M NC N NC ) → (q NC Tc N = (M +c q) → p NC M = Tc p))
693, 68sylbid 206 . 2 ((M NC N NC ) → (Mc Tc Np NC M = Tc p))
70693impia 1148 1 ((M NC N NC Mc Tc N) → p NC M = Tc p)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  ℘1cpw1 4135   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089   Nc cnc 6091   Tc ctc 6093 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-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  df-tc 6103 This theorem is referenced by:  ce2le  6233  nchoicelem19  6307
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