Theorem tfin11 4494

Description: The finite T operator is one-to-one over the naturals. Theorem X.1.30 of [Rosser] p. 528. (Contributed by SF, 24-Jan-2015.)

Assertion

Ref	Expression
tfin11	⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N)

Proof of Theorem tfin11

Dummy variables a b p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinnnul 4491	. . . . . . . 8 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → Tfin M ≠ ∅)
2	1	ex 423	. . . . . . 7 ⊢ (M ∈ Nn → (M ≠ ∅ → Tfin M ≠ ∅))
3	2	necon4d 2580	. . . . . 6 ⊢ (M ∈ Nn → ( Tfin M = ∅ → M = ∅))
4	3	3ad2ant1 976	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M = ∅ → M = ∅))
5	4	impcom 419	. . . 4 ⊢ (( Tfin M = ∅ ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → M = ∅)
6		eqeq1 2359	. . . . . . . 8 ⊢ ( Tfin M = Tfin N → ( Tfin M = ∅ ↔ Tfin N = ∅))
7	6	adantl 452	. . . . . . 7 ⊢ ((N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M = ∅ ↔ Tfin N = ∅))
8		tfinnnul 4491	. . . . . . . . . 10 ⊢ ((N ∈ Nn ∧ N ≠ ∅) → Tfin N ≠ ∅)
9	8	ex 423	. . . . . . . . 9 ⊢ (N ∈ Nn → (N ≠ ∅ → Tfin N ≠ ∅))
10	9	necon4d 2580	. . . . . . . 8 ⊢ (N ∈ Nn → ( Tfin N = ∅ → N = ∅))
11	10	adantr 451	. . . . . . 7 ⊢ ((N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin N = ∅ → N = ∅))
12	7, 11	sylbid 206	. . . . . 6 ⊢ ((N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M = ∅ → N = ∅))
13	12	3adant1 973	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M = ∅ → N = ∅))
14	13	impcom 419	. . . 4 ⊢ (( Tfin M = ∅ ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → N = ∅)
15	5, 14	eqtr4d 2388	. . 3 ⊢ (( Tfin M = ∅ ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → M = N)
16	15	ex 423	. 2 ⊢ ( Tfin M = ∅ → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N))
17		neeq1 2525	. . . . . . . 8 ⊢ ( Tfin M = Tfin N → ( Tfin M ≠ ∅ ↔ Tfin N ≠ ∅))
18	17	biimpd 198	. . . . . . 7 ⊢ ( Tfin M = Tfin N → ( Tfin M ≠ ∅ → Tfin N ≠ ∅))
19	18	ancld 536	. . . . . 6 ⊢ ( Tfin M = Tfin N → ( Tfin M ≠ ∅ → ( Tfin M ≠ ∅ ∧ Tfin N ≠ ∅)))
20		tfineq 4489	. . . . . . . . 9 ⊢ (M = ∅ → Tfin M = Tfin ∅)
21		tfinnul 4492	. . . . . . . . 9 ⊢ Tfin ∅ = ∅
22	20, 21	syl6eq 2401	. . . . . . . 8 ⊢ (M = ∅ → Tfin M = ∅)
23	22	necon3i 2556	. . . . . . 7 ⊢ ( Tfin M ≠ ∅ → M ≠ ∅)
24		tfineq 4489	. . . . . . . . 9 ⊢ (N = ∅ → Tfin N = Tfin ∅)
25	24, 21	syl6eq 2401	. . . . . . . 8 ⊢ (N = ∅ → Tfin N = ∅)
26	25	necon3i 2556	. . . . . . 7 ⊢ ( Tfin N ≠ ∅ → N ≠ ∅)
27	23, 26	anim12i 549	. . . . . 6 ⊢ (( Tfin M ≠ ∅ ∧ Tfin N ≠ ∅) → (M ≠ ∅ ∧ N ≠ ∅))
28	19, 27	syl6 29	. . . . 5 ⊢ ( Tfin M = Tfin N → ( Tfin M ≠ ∅ → (M ≠ ∅ ∧ N ≠ ∅)))
29	28	3ad2ant3 978	. . . 4 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M ≠ ∅ → (M ≠ ∅ ∧ N ≠ ∅)))
30		tfinprop 4490	. . . . . . 7 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M))
31	30	ex 423	. . . . . 6 ⊢ (M ∈ Nn → (M ≠ ∅ → ( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M)))
32	31	3ad2ant1 976	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → (M ≠ ∅ → ( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M)))
33		tfinprop 4490	. . . . . . 7 ⊢ ((N ∈ Nn ∧ N ≠ ∅) → ( Tfin N ∈ Nn ∧ ∃b ∈ N ℘1b ∈ Tfin N))
34	33	ex 423	. . . . . 6 ⊢ (N ∈ Nn → (N ≠ ∅ → ( Tfin N ∈ Nn ∧ ∃b ∈ N ℘1b ∈ Tfin N)))
35	34	3ad2ant2 977	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → (N ≠ ∅ → ( Tfin N ∈ Nn ∧ ∃b ∈ N ℘1b ∈ Tfin N)))
36		reeanv 2779	. . . . . . . 8 ⊢ (∃a ∈ M ∃b ∈ N (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ↔ (∃a ∈ M ℘1a ∈ Tfin M ∧ ∃b ∈ N ℘1b ∈ Tfin N))
37		simp31 991	. . . . . . . . . . . . 13 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → M ∈ Nn )
38		tfincl 4493	. . . . . . . . . . . . 13 ⊢ (M ∈ Nn → Tfin M ∈ Nn )
39	37, 38	syl 15	. . . . . . . . . . . 12 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → Tfin M ∈ Nn )
40		simp2l 981	. . . . . . . . . . . 12 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → ℘1a ∈ Tfin M)
41		simp2r 982	. . . . . . . . . . . . 13 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → ℘1b ∈ Tfin N)
42		simp33 993	. . . . . . . . . . . . 13 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → Tfin M = Tfin N)
43	41, 42	eleqtrrd 2430	. . . . . . . . . . . 12 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → ℘1b ∈ Tfin M)
44		ncfinlower 4484	. . . . . . . . . . . 12 ⊢ (( Tfin M ∈ Nn ∧ ℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin M) → ∃p ∈ Nn (a ∈ p ∧ b ∈ p))
45	39, 40, 43, 44	syl3anc 1182	. . . . . . . . . . 11 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → ∃p ∈ Nn (a ∈ p ∧ b ∈ p))
46		simpl31 1036	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → M ∈ Nn )
47		simprl 732	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → p ∈ Nn )
48		simpl1l 1006	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → a ∈ M)
49		simprrl 740	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → a ∈ p)
50		nnceleq 4431	. . . . . . . . . . . . . . 15 ⊢ (((M ∈ Nn ∧ p ∈ Nn ) ∧ (a ∈ M ∧ a ∈ p)) → M = p)
51	46, 47, 48, 49, 50	syl22anc 1183	. . . . . . . . . . . . . 14 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → M = p)
52		simpl32 1037	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → N ∈ Nn )
53		simpl1r 1007	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → b ∈ N)
54		simprrr 741	. . . . . . . . . . . . . . 15 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → b ∈ p)
55		nnceleq 4431	. . . . . . . . . . . . . . 15 ⊢ (((N ∈ Nn ∧ p ∈ Nn ) ∧ (b ∈ N ∧ b ∈ p)) → N = p)
56	52, 47, 53, 54, 55	syl22anc 1183	. . . . . . . . . . . . . 14 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → N = p)
57	51, 56	eqtr4d 2388	. . . . . . . . . . . . 13 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ (p ∈ Nn ∧ (a ∈ p ∧ b ∈ p))) → M = N)
58	57	expr 598	. . . . . . . . . . . 12 ⊢ ((((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) ∧ p ∈ Nn ) → ((a ∈ p ∧ b ∈ p) → M = N))
59	58	rexlimdva 2739	. . . . . . . . . . 11 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → (∃p ∈ Nn (a ∈ p ∧ b ∈ p) → M = N))
60	45, 59	mpd 14	. . . . . . . . . 10 ⊢ (((a ∈ M ∧ b ∈ N) ∧ (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) ∧ (M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N)) → M = N)
61	60	3exp 1150	. . . . . . . . 9 ⊢ ((a ∈ M ∧ b ∈ N) → ((℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N)))
62	61	rexlimivv 2744	. . . . . . . 8 ⊢ (∃a ∈ M ∃b ∈ N (℘1a ∈ Tfin M ∧ ℘1b ∈ Tfin N) → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N))
63	36, 62	sylbir 204	. . . . . . 7 ⊢ ((∃a ∈ M ℘1a ∈ Tfin M ∧ ∃b ∈ N ℘1b ∈ Tfin N) → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N))
64	63	ad2ant2l 726	. . . . . 6 ⊢ ((( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M) ∧ ( Tfin N ∈ Nn ∧ ∃b ∈ N ℘1b ∈ Tfin N)) → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N))
65	64	com12 27	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ((( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M) ∧ ( Tfin N ∈ Nn ∧ ∃b ∈ N ℘1b ∈ Tfin N)) → M = N))
66	32, 35, 65	syl2and 469	. . . 4 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ((M ≠ ∅ ∧ N ≠ ∅) → M = N))
67	29, 66	syld 40	. . 3 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → ( Tfin M ≠ ∅ → M = N))
68	67	com12 27	. 2 ⊢ ( Tfin M ≠ ∅ → ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N))
69	16, 68	pm2.61ine 2593	1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ Tfin M = Tfin N) → M = N)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅c0 3551  ℘₁cpw1 4136   Nn cnnc 4374   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-13 1712  ax-14 1714  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-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-tfin 4444

This theorem is used by:  tfinnn  4535  vfinspsslem1  4551