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Theorem tfinsuc 4498
 Description: The finite T operator over a successor. (Contributed by SF, 30-Jan-2015.)
Assertion
Ref Expression
tfinsuc ((A Nn (A +c 1c) ≠ ) → Tfin (A +c 1c) = ( Tfin A +c 1c))

Proof of Theorem tfinsuc
Dummy variables a b x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3559 . . 3 ((A +c 1c) ≠ a a (A +c 1c))
2 peano2 4403 . . . . . . . 8 (A Nn → (A +c 1c) Nn )
3 tfincl 4492 . . . . . . . 8 ((A +c 1c) NnTfin (A +c 1c) Nn )
42, 3syl 15 . . . . . . 7 (A NnTfin (A +c 1c) Nn )
54adantr 451 . . . . . 6 ((A Nn a (A +c 1c)) → Tfin (A +c 1c) Nn )
6 tfincl 4492 . . . . . . . 8 (A NnTfin A Nn )
7 peano2 4403 . . . . . . . 8 ( Tfin A Nn → ( Tfin A +c 1c) Nn )
86, 7syl 15 . . . . . . 7 (A Nn → ( Tfin A +c 1c) Nn )
98adantr 451 . . . . . 6 ((A Nn a (A +c 1c)) → ( Tfin A +c 1c) Nn )
10 tfinpw1 4494 . . . . . . 7 (((A +c 1c) Nn a (A +c 1c)) → 1a Tfin (A +c 1c))
112, 10sylan 457 . . . . . 6 ((A Nn a (A +c 1c)) → 1a Tfin (A +c 1c))
12 elsuc 4413 . . . . . . . 8 (a (A +c 1c) ↔ b A x ba = (b ∪ {x}))
13 tfinpw1 4494 . . . . . . . . . . . 12 ((A Nn b A) → 1b Tfin A)
1413adantrr 697 . . . . . . . . . . 11 ((A Nn (b A x b)) → 1b Tfin A)
15 vex 2862 . . . . . . . . . . . . . . 15 x V
1615elcompl 3225 . . . . . . . . . . . . . 14 (x b ↔ ¬ x b)
17 snelpw1 4146 . . . . . . . . . . . . . 14 ({x} 1bx b)
1816, 17xchbinxr 302 . . . . . . . . . . . . 13 (x b ↔ ¬ {x} 1b)
1918biimpi 186 . . . . . . . . . . . 12 (x b → ¬ {x} 1b)
2019ad2antll 709 . . . . . . . . . . 11 ((A Nn (b A x b)) → ¬ {x} 1b)
21 snex 4111 . . . . . . . . . . . 12 {x} V
2221elsuci 4414 . . . . . . . . . . 11 ((1b Tfin A ¬ {x} 1b) → (1b ∪ {{x}}) ( Tfin A +c 1c))
2314, 20, 22syl2anc 642 . . . . . . . . . 10 ((A Nn (b A x b)) → (1b ∪ {{x}}) ( Tfin A +c 1c))
24 pw1eq 4143 . . . . . . . . . . . 12 (a = (b ∪ {x}) → 1a = 1(b ∪ {x}))
25 pw1un 4163 . . . . . . . . . . . . 13 1(b ∪ {x}) = (1b1{x})
2615pw1sn 4165 . . . . . . . . . . . . . 14 1{x} = {{x}}
2726uneq2i 3415 . . . . . . . . . . . . 13 (1b1{x}) = (1b ∪ {{x}})
2825, 27eqtri 2373 . . . . . . . . . . . 12 1(b ∪ {x}) = (1b ∪ {{x}})
2924, 28syl6eq 2401 . . . . . . . . . . 11 (a = (b ∪ {x}) → 1a = (1b ∪ {{x}}))
3029eleq1d 2419 . . . . . . . . . 10 (a = (b ∪ {x}) → (1a ( Tfin A +c 1c) ↔ (1b ∪ {{x}}) ( Tfin A +c 1c)))
3123, 30syl5ibrcom 213 . . . . . . . . 9 ((A Nn (b A x b)) → (a = (b ∪ {x}) → 1a ( Tfin A +c 1c)))
3231rexlimdvva 2745 . . . . . . . 8 (A Nn → (b A x ba = (b ∪ {x}) → 1a ( Tfin A +c 1c)))
3312, 32syl5bi 208 . . . . . . 7 (A Nn → (a (A +c 1c) → 1a ( Tfin A +c 1c)))
3433imp 418 . . . . . 6 ((A Nn a (A +c 1c)) → 1a ( Tfin A +c 1c))
35 nnceleq 4430 . . . . . 6 ((( Tfin (A +c 1c) Nn ( Tfin A +c 1c) Nn ) (1a Tfin (A +c 1c) 1a ( Tfin A +c 1c))) → Tfin (A +c 1c) = ( Tfin A +c 1c))
365, 9, 11, 34, 35syl22anc 1183 . . . . 5 ((A Nn a (A +c 1c)) → Tfin (A +c 1c) = ( Tfin A +c 1c))
3736ex 423 . . . 4 (A Nn → (a (A +c 1c) → Tfin (A +c 1c) = ( Tfin A +c 1c)))
3837exlimdv 1636 . . 3 (A Nn → (a a (A +c 1c) → Tfin (A +c 1c) = ( Tfin A +c 1c)))
391, 38syl5bi 208 . 2 (A Nn → ((A +c 1c) ≠ Tfin (A +c 1c) = ( Tfin A +c 1c)))
4039imp 418 1 ((A Nn (A +c 1c) ≠ ) → Tfin (A +c 1c) = ( Tfin A +c 1c))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207  ∅c0 3550  {csn 3737  1cc1c 4134  ℘1cpw1 4135   Nn cnnc 4373   +c cplc 4375   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:  tfin1c  4499  tfinltfinlem1  4500  sfintfin  4532  tfinnn  4534
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