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Theorem tfinlefin 4502
Description: Ordering rule for the finite T operation. Theorem X.1.33 of [Rosser] p. 529. (Contributed by SF, 2-Feb-2015.)
Assertion
Ref Expression
tfinlefin ((M Nn N Nn ) → (⟪M, Nfin ↔ ⟪ Tfin M, Tfin Nfin ))

Proof of Theorem tfinlefin
StepHypRef Expression
1 tfinltfin 4501 . . . 4 ((N Nn M Nn ) → (⟪N, M <fin ↔ ⟪ Tfin N, Tfin M <fin ))
21ancoms 439 . . 3 ((M Nn N Nn ) → (⟪N, M <fin ↔ ⟪ Tfin N, Tfin M <fin ))
32notbid 285 . 2 ((M Nn N Nn ) → (¬ ⟪N, M <fin ↔ ¬ ⟪ Tfin N, Tfin M <fin ))
4 lenltfin 4469 . 2 ((M Nn N Nn ) → (⟪M, Nfin ↔ ¬ ⟪N, M <fin ))
5 tfincl 4492 . . 3 (M NnTfin M Nn )
6 tfincl 4492 . . 3 (N NnTfin N Nn )
7 lenltfin 4469 . . 3 (( Tfin M Nn Tfin N Nn ) → (⟪ Tfin M, Tfin Nfin ↔ ¬ ⟪ Tfin N, Tfin M <fin ))
85, 6, 7syl2an 463 . 2 ((M Nn N Nn ) → (⟪ Tfin M, Tfin Nfin ↔ ¬ ⟪ Tfin N, Tfin M <fin ))
93, 4, 83bitr4d 276 1 ((M Nn N Nn ) → (⟪M, Nfin ↔ ⟪ Tfin M, Tfin Nfin ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wa 358   wcel 1710  copk 4057   Nn cnnc 4373  fin clefin 4432   <fin cltfin 4433   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-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-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-lefin 4440  df-ltfin 4441  df-tfin 4443
This theorem is referenced by:  vfintle  4546
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