Theorem tfinltfin 4502

Description: Ordering rule for the finite T operation. Corollary to theorem X.1.33 of [Rosser] p. 529. (Contributed by SF, 1-Feb-2015.)

Assertion

Ref	Expression
tfinltfin	⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ <fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ <fin ))

Proof of Theorem tfinltfin

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinltfinlem1 4501	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ <fin → ⟪ Tfin M, Tfin N⟫ ∈ <fin ))
2		tfineq 4489	. . . . . . . . 9 ⊢ (M = ∅ → Tfin M = Tfin ∅)
3		tfinnul 4492	. . . . . . . . 9 ⊢ Tfin ∅ = ∅
4	2, 3	syl6eq 2401	. . . . . . . 8 ⊢ (M = ∅ → Tfin M = ∅)
5		df-ne 2519	. . . . . . . . 9 ⊢ ( Tfin M ≠ ∅ ↔ ¬ Tfin M = ∅)
6	5	con2bii 322	. . . . . . . 8 ⊢ ( Tfin M = ∅ ↔ ¬ Tfin M ≠ ∅)
7	4, 6	sylib 188	. . . . . . 7 ⊢ (M = ∅ → ¬ Tfin M ≠ ∅)
8	7	intnanrd 883	. . . . . 6 ⊢ (M = ∅ → ¬ ( Tfin M ≠ ∅ ∧ ∃x ∈ Nn Tfin N = (( Tfin M +c x) +c 1c)))
9		tfinex 4486	. . . . . . 7 ⊢ Tfin M ∈ V
10		tfinex 4486	. . . . . . 7 ⊢ Tfin N ∈ V
11		opkltfing 4450	. . . . . . 7 ⊢ (( Tfin M ∈ V ∧ Tfin N ∈ V) → (⟪ Tfin M, Tfin N⟫ ∈ <fin ↔ ( Tfin M ≠ ∅ ∧ ∃x ∈ Nn Tfin N = (( Tfin M +c x) +c 1c))))
12	9, 10, 11	mp2an 653	. . . . . 6 ⊢ (⟪ Tfin M, Tfin N⟫ ∈ <fin ↔ ( Tfin M ≠ ∅ ∧ ∃x ∈ Nn Tfin N = (( Tfin M +c x) +c 1c)))
13	8, 12	sylnibr 296	. . . . 5 ⊢ (M = ∅ → ¬ ⟪ Tfin M, Tfin N⟫ ∈ <fin )
14	13	pm2.21d 98	. . . 4 ⊢ (M = ∅ → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin ))
15	14	a1d 22	. . 3 ⊢ (M = ∅ → ((M ∈ Nn ∧ N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin )))
16		tfinprop 4490	. . . . . . . . . . . . . 14 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ( Tfin M ∈ Nn ∧ ∃y ∈ M ℘1y ∈ Tfin M))
17	16	simpld 445	. . . . . . . . . . . . 13 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → Tfin M ∈ Nn )
18		ltfinirr 4458	. . . . . . . . . . . . 13 ⊢ ( Tfin M ∈ Nn → ¬ ⟪ Tfin M, Tfin M⟫ ∈ <fin )
19	17, 18	syl 15	. . . . . . . . . . . 12 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ¬ ⟪ Tfin M, Tfin M⟫ ∈ <fin )
20	19	3adant2 974	. . . . . . . . . . 11 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → ¬ ⟪ Tfin M, Tfin M⟫ ∈ <fin )
21		opkeq2 4061	. . . . . . . . . . . . 13 ⊢ ( Tfin M = Tfin N → ⟪ Tfin M, Tfin M⟫ = ⟪ Tfin M, Tfin N⟫)
22	21	eleq1d 2419	. . . . . . . . . . . 12 ⊢ ( Tfin M = Tfin N → (⟪ Tfin M, Tfin M⟫ ∈ <fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ <fin ))
23	22	notbid 285	. . . . . . . . . . 11 ⊢ ( Tfin M = Tfin N → (¬ ⟪ Tfin M, Tfin M⟫ ∈ <fin ↔ ¬ ⟪ Tfin M, Tfin N⟫ ∈ <fin ))
24	20, 23	syl5ibcom 211	. . . . . . . . . 10 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → ( Tfin M = Tfin N → ¬ ⟪ Tfin M, Tfin N⟫ ∈ <fin ))
25	24	con2d 107	. . . . . . . . 9 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ¬ Tfin M = Tfin N))
26	25	imp 418	. . . . . . . 8 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → ¬ Tfin M = Tfin N)
27		tfineq 4489	. . . . . . . 8 ⊢ (M = N → Tfin M = Tfin N)
28	26, 27	nsyl 113	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → ¬ M = N)
29		simpl1 958	. . . . . . . . . . . . 13 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → M ∈ Nn )
30		simpl3 960	. . . . . . . . . . . . 13 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → M ≠ ∅)
31	29, 30, 17	syl2anc 642	. . . . . . . . . . . 12 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → Tfin M ∈ Nn )
32		simpl2 959	. . . . . . . . . . . . 13 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → N ∈ Nn )
33		simprr 733	. . . . . . . . . . . . 13 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → N ≠ ∅)
34		tfinprop 4490	. . . . . . . . . . . . . 14 ⊢ ((N ∈ Nn ∧ N ≠ ∅) → ( Tfin N ∈ Nn ∧ ∃y ∈ N ℘1y ∈ Tfin N))
35	34	simpld 445	. . . . . . . . . . . . 13 ⊢ ((N ∈ Nn ∧ N ≠ ∅) → Tfin N ∈ Nn )
36	32, 33, 35	syl2anc 642	. . . . . . . . . . . 12 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → Tfin N ∈ Nn )
37	31, 36	jca 518	. . . . . . . . . . 11 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → ( Tfin M ∈ Nn ∧ Tfin N ∈ Nn ))
38		simprl 732	. . . . . . . . . . 11 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → ⟪ Tfin M, Tfin N⟫ ∈ <fin )
39		ltfinasym 4461	. . . . . . . . . . 11 ⊢ (( Tfin M ∈ Nn ∧ Tfin N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
40	37, 38, 39	sylc 56	. . . . . . . . . 10 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ N ≠ ∅)) → ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin )
41	40	expr 598	. . . . . . . . 9 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → (N ≠ ∅ → ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
42		imnan 411	. . . . . . . . 9 ⊢ ((N ≠ ∅ → ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin ) ↔ ¬ (N ≠ ∅ ∧ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
43	41, 42	sylib 188	. . . . . . . 8 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → ¬ (N ≠ ∅ ∧ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
44		opkltfing 4450	. . . . . . . . . . . . . 14 ⊢ ((N ∈ Nn ∧ M ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ (N ≠ ∅ ∧ ∃y ∈ Nn M = ((N +c y) +c 1c))))
45	44	ancoms 439	. . . . . . . . . . . . 13 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ (N ≠ ∅ ∧ ∃y ∈ Nn M = ((N +c y) +c 1c))))
46	45	3adant3 975	. . . . . . . . . . . 12 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → (⟪N, M⟫ ∈ <fin ↔ (N ≠ ∅ ∧ ∃y ∈ Nn M = ((N +c y) +c 1c))))
47	46	simprbda 606	. . . . . . . . . . 11 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪N, M⟫ ∈ <fin ) → N ≠ ∅)
48	47	adantrl 696	. . . . . . . . . 10 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → N ≠ ∅)
49		simpl2 959	. . . . . . . . . . . 12 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → N ∈ Nn )
50		simpl1 958	. . . . . . . . . . . 12 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → M ∈ Nn )
51	49, 50	jca 518	. . . . . . . . . . 11 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → (N ∈ Nn ∧ M ∈ Nn ))
52		simprr 733	. . . . . . . . . . 11 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → ⟪N, M⟫ ∈ <fin )
53		tfinltfinlem1 4501	. . . . . . . . . . 11 ⊢ ((N ∈ Nn ∧ M ∈ Nn ) → (⟪N, M⟫ ∈ <fin → ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
54	51, 52, 53	sylc 56	. . . . . . . . . 10 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → ⟪ Tfin N, Tfin M⟫ ∈ <fin )
55	48, 54	jca 518	. . . . . . . . 9 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ (⟪ Tfin M, Tfin N⟫ ∈ <fin ∧ ⟪N, M⟫ ∈ <fin )) → (N ≠ ∅ ∧ ⟪ Tfin N, Tfin M⟫ ∈ <fin ))
56	55	expr 598	. . . . . . . 8 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → (⟪N, M⟫ ∈ <fin → (N ≠ ∅ ∧ ⟪ Tfin N, Tfin M⟫ ∈ <fin )))
57	43, 56	mtod 168	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → ¬ ⟪N, M⟫ ∈ <fin )
58		ltfintri 4467	. . . . . . . 8 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → (⟪M, N⟫ ∈ <fin ∨ M = N ∨ ⟪N, M⟫ ∈ <fin ))
59	58	adantr 451	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → (⟪M, N⟫ ∈ <fin ∨ M = N ∨ ⟪N, M⟫ ∈ <fin ))
60	28, 57, 59	ecase23d 1285	. . . . . 6 ⊢ (((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) ∧ ⟪ Tfin M, Tfin N⟫ ∈ <fin ) → ⟪M, N⟫ ∈ <fin )
61	60	ex 423	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin ))
62	61	3expa 1151	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ M ≠ ∅) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin ))
63	62	expcom 424	. . 3 ⊢ (M ≠ ∅ → ((M ∈ Nn ∧ N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin )))
64	15, 63	pm2.61ine 2593	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ <fin → ⟪M, N⟫ ∈ <fin ))
65	1, 64	impbid 183	1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ <fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ <fin ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∨ w3o 933   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  Vcvv 2860  ∅c0 3551  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136   Nn cnnc 4374   +_c cplc 4376   <_fin cltfin 4434   T_fin ctfin 4436

This proof depends on 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 proof 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 used by:  tfinlefin  4503  sfinltfin  4536