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Theorem tfincl 4492
 Description: Closure law for finite T operation. (Contributed by SF, 2-Feb-2015.)
Assertion
Ref Expression
tfincl (N NnTfin N Nn )

Proof of Theorem tfincl
Dummy variable a is distinct from all other variables.
StepHypRef Expression
1 tfinnul 4491 . . . . 5 Tfin =
2 tfineq 4488 . . . . 5 (N = Tfin N = Tfin )
3 id 19 . . . . 5 (N = N = )
41, 2, 33eqtr4a 2411 . . . 4 (N = Tfin N = N)
54eleq1d 2419 . . 3 (N = → ( Tfin N NnN Nn ))
65biimprd 214 . 2 (N = → (N NnTfin N Nn ))
7 tfinprop 4489 . . . 4 ((N Nn N) → ( Tfin N Nn a N 1a Tfin N))
87simpld 445 . . 3 ((N Nn N) → Tfin N Nn )
98expcom 424 . 2 (N → (N NnTfin N Nn ))
106, 9pm2.61ine 2592 1 (N NnTfin N Nn )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ℘1cpw1 4135   Nn cnnc 4373   Tfin ctfin 4435 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-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-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-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-tfin 4443 This theorem is referenced by:  tfin11  4493  tfinpw1  4494  ncfintfin  4495  tfindi  4496  tfin0c  4497  tfinsuc  4498  tfinlefin  4502  vfinspsslem1  4550  vfinncsp  4554
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