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Theorem tfin11 4493
 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.
StepHypRef Expression
1 tfinnnul 4490 . . . . . . . 8 ((M Nn M) → Tfin M)
21ex 423 . . . . . . 7 (M Nn → (MTfin M))
32necon4d 2579 . . . . . 6 (M Nn → ( Tfin M = M = ))
433ad2ant1 976 . . . . 5 ((M Nn N Nn Tfin M = Tfin N) → ( Tfin M = M = ))
54impcom 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 = ))
76adantl 452 . . . . . . 7 ((N Nn Tfin M = Tfin N) → ( Tfin M = Tfin N = ))
8 tfinnnul 4490 . . . . . . . . . 10 ((N Nn N) → Tfin N)
98ex 423 . . . . . . . . 9 (N Nn → (NTfin N))
109necon4d 2579 . . . . . . . 8 (N Nn → ( Tfin N = N = ))
1110adantr 451 . . . . . . 7 ((N Nn Tfin M = Tfin N) → ( Tfin N = N = ))
127, 11sylbid 206 . . . . . 6 ((N Nn Tfin M = Tfin N) → ( Tfin M = N = ))
13123adant1 973 . . . . 5 ((M Nn N Nn Tfin M = Tfin N) → ( Tfin M = N = ))
1413impcom 419 . . . 4 (( Tfin M = (M Nn N Nn Tfin M = Tfin N)) → N = )
155, 14eqtr4d 2388 . . 3 (( Tfin M = (M Nn N Nn Tfin M = Tfin N)) → M = N)
1615ex 423 . 2 ( Tfin M = → ((M Nn N Nn Tfin M = Tfin N) → M = N))
17 neeq1 2524 . . . . . . . 8 ( Tfin M = Tfin N → ( Tfin MTfin N))
1817biimpd 198 . . . . . . 7 ( Tfin M = Tfin N → ( Tfin MTfin N))
1918ancld 536 . . . . . 6 ( Tfin M = Tfin N → ( Tfin M → ( Tfin M Tfin N)))
20 tfineq 4488 . . . . . . . . 9 (M = Tfin M = Tfin )
21 tfinnul 4491 . . . . . . . . 9 Tfin =
2220, 21syl6eq 2401 . . . . . . . 8 (M = Tfin M = )
2322necon3i 2555 . . . . . . 7 ( Tfin MM)
24 tfineq 4488 . . . . . . . . 9 (N = Tfin N = Tfin )
2524, 21syl6eq 2401 . . . . . . . 8 (N = Tfin N = )
2625necon3i 2555 . . . . . . 7 ( Tfin NN)
2723, 26anim12i 549 . . . . . 6 (( Tfin M Tfin N) → (M N))
2819, 27syl6 29 . . . . 5 ( Tfin M = Tfin N → ( Tfin M → (M N)))
29283ad2ant3 978 . . . 4 ((M Nn N Nn Tfin M = Tfin N) → ( Tfin M → (M N)))
30 tfinprop 4489 . . . . . . 7 ((M Nn M) → ( Tfin M Nn a M 1a Tfin M))
3130ex 423 . . . . . 6 (M Nn → (M → ( Tfin M Nn a M 1a Tfin M)))
32313ad2ant1 976 . . . . 5 ((M Nn N Nn Tfin M = Tfin N) → (M → ( Tfin M Nn a M 1a Tfin M)))
33 tfinprop 4489 . . . . . . 7 ((N Nn N) → ( Tfin N Nn b N 1b Tfin N))
3433ex 423 . . . . . 6 (N Nn → (N → ( Tfin N Nn b N 1b Tfin N)))
35343ad2ant2 977 . . . . 5 ((M Nn N Nn Tfin M = Tfin N) → (N → ( Tfin N Nn b N 1b Tfin N)))
36 reeanv 2778 . . . . . . . 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 4492 . . . . . . . . . . . . 13 (M NnTfin M Nn )
3937, 38syl 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)
4341, 42eleqtrrd 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 4483 . . . . . . . . . . . 12 (( Tfin M Nn 1a Tfin M 1b Tfin M) → p Nn (a p b p))
4539, 40, 43, 44syl3anc 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 4430 . . . . . . . . . . . . . . 15 (((M Nn p Nn ) (a M a p)) → M = p)
5146, 47, 48, 49, 50syl22anc 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 4430 . . . . . . . . . . . . . . 15 (((N Nn p Nn ) (b N b p)) → N = p)
5652, 47, 53, 54, 55syl22anc 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)
5751, 56eqtr4d 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)
5857expr 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))
5958rexlimdva 2738 . . . . . . . . . . 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))
6045, 59mpd 14 . . . . . . . . . 10 (((a M b N) (1a Tfin M 1b Tfin N) (M Nn N Nn Tfin M = Tfin N)) → M = N)
61603exp 1150 . . . . . . . . 9 ((a M b N) → ((1a Tfin M 1b Tfin N) → ((M Nn N Nn Tfin M = Tfin N) → M = N)))
6261rexlimivv 2743 . . . . . . . 8 (a M b N (1a Tfin M 1b Tfin N) → ((M Nn N Nn Tfin M = Tfin N) → M = N))
6336, 62sylbir 204 . . . . . . 7 ((a M 1a Tfin M b N 1b Tfin N) → ((M Nn N Nn Tfin M = Tfin N) → M = N))
6463ad2ant2l 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))
6564com12 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))
6632, 35, 65syl2and 469 . . . 4 ((M Nn N Nn Tfin M = Tfin N) → ((M N) → M = N))
6729, 66syld 40 . . 3 ((M Nn N Nn Tfin M = Tfin N) → ( Tfin MM = N))
6867com12 27 . 2 ( Tfin M → ((M Nn N Nn Tfin M = Tfin N) → M = N))
6916, 68pm2.61ine 2592 1 ((M Nn N Nn Tfin M = Tfin N) → M = N)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ℘1cpw1 4135   Nn cnnc 4373   Tfin ctfin 4435 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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:  tfinnn  4534  vfinspsslem1  4550
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