Theorem tfinlefin 4503

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, N⟫ ∈ ≤fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ ≤fin ))

Proof of Theorem tfinlefin

Step	Hyp	Ref	Expression
1		tfinltfin 4502	. . . 4 ⊢ ((N ∈ Nn ∧ M ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
2	1	ancoms 439	. . 3 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
3	2	notbid 285	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (¬ ⟪N, M⟫ ∈ <fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
4		lenltfin 4470	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ ≤fin ↔ ¬ ⟪N, M⟫ ∈ <fin ))
5		tfincl 4493	. . 3 ⊢ (M ∈ Nn → Tfin M ∈ Nn )
6		tfincl 4493	. . 3 ⊢ (N ∈ Nn → Tfin N ∈ Nn )
7		lenltfin 4470	. . 3 ⊢ (( Tfin M ∈ Nn ∧ Tfin N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ ≤fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
8	5, 6, 7	syl2an 463	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ ≤fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
9	3, 4, 8	3bitr4d 276	1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ ≤fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ ≤fin ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ⟪copk 4058   Nn cnnc 4374   ≤_fin clefin 4433   <_fin cltfin 4434   T_fin ctfin 4436

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-lefin 4441  df-ltfin 4442  df-tfin 4444

This theorem is referenced by:  vfintle  4547