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Theorem sfin111 4536
Description: The finite smaller relationship is one-to-one in its first argument. Theorem X.1.48 of [Rosser] p. 533. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
sfin111 (( Sfin (M, P) Sfin (N, P)) → M = N)

Proof of Theorem sfin111
Dummy variable a is distinct from all other variables.
StepHypRef Expression
1 df-sfin 4446 . . . . . . 7 ( Sfin (N, P) ↔ (N Nn P Nn a(1a N a P)))
21simp2bi 971 . . . . . 6 ( Sfin (N, P) → P Nn )
32adantl 452 . . . . 5 (( Sfin (M, P) Sfin (N, P)) → P Nn )
4 ltfinirr 4457 . . . . 5 (P Nn → ¬ ⟪P, P <fin )
53, 4syl 15 . . . 4 (( Sfin (M, P) Sfin (N, P)) → ¬ ⟪P, P <fin )
6 sfinltfin 4535 . . . 4 ((( Sfin (M, P) Sfin (N, P)) M, N <fin ) → ⟪P, P <fin )
75, 6mtand 640 . . 3 (( Sfin (M, P) Sfin (N, P)) → ¬ ⟪M, N <fin )
8 sfinltfin 4535 . . . . . 6 ((( Sfin (N, P) Sfin (M, P)) N, M <fin ) → ⟪P, P <fin )
98ex 423 . . . . 5 (( Sfin (N, P) Sfin (M, P)) → (⟪N, M <fin → ⟪P, P <fin ))
109ancoms 439 . . . 4 (( Sfin (M, P) Sfin (N, P)) → (⟪N, M <fin → ⟪P, P <fin ))
115, 10mtod 168 . . 3 (( Sfin (M, P) Sfin (N, P)) → ¬ ⟪N, M <fin )
12 ioran 476 . . 3 (¬ (⟪M, N <fin N, M <fin ) ↔ (¬ ⟪M, N <fin ¬ ⟪N, M <fin ))
137, 11, 12sylanbrc 645 . 2 (( Sfin (M, P) Sfin (N, P)) → ¬ (⟪M, N <fin N, M <fin ))
14 df-sfin 4446 . . . . . . 7 ( Sfin (M, P) ↔ (M Nn P Nn a(1a M a P)))
1514simp1bi 970 . . . . . 6 ( Sfin (M, P) → M Nn )
1615adantr 451 . . . . 5 (( Sfin (M, P) Sfin (N, P)) → M Nn )
171simp1bi 970 . . . . . 6 ( Sfin (N, P) → N Nn )
1817adantl 452 . . . . 5 (( Sfin (M, P) Sfin (N, P)) → N Nn )
19 ne0i 3556 . . . . . . . . . 10 (1a MM)
2019adantr 451 . . . . . . . . 9 ((1a M a P) → M)
2120exlimiv 1634 . . . . . . . 8 (a(1a M a P) → M)
22213ad2ant3 978 . . . . . . 7 ((M Nn P Nn a(1a M a P)) → M)
2314, 22sylbi 187 . . . . . 6 ( Sfin (M, P) → M)
2423adantr 451 . . . . 5 (( Sfin (M, P) Sfin (N, P)) → M)
25 ltfintri 4466 . . . . 5 ((M Nn N Nn M) → (⟪M, N <fin M = N N, M <fin ))
2616, 18, 24, 25syl3anc 1182 . . . 4 (( Sfin (M, P) Sfin (N, P)) → (⟪M, N <fin M = N N, M <fin ))
27 df-3or 935 . . . 4 ((⟪M, N <fin M = N N, M <fin ) ↔ ((⟪M, N <fin M = N) N, M <fin ))
2826, 27sylib 188 . . 3 (( Sfin (M, P) Sfin (N, P)) → ((⟪M, N <fin M = N) N, M <fin ))
29 or32 513 . . 3 (((⟪M, N <fin M = N) N, M <fin ) ↔ ((⟪M, N <fin N, M <fin ) M = N))
3028, 29sylib 188 . 2 (( Sfin (M, P) Sfin (N, P)) → ((⟪M, N <fin N, M <fin ) M = N))
31 orel1 371 . 2 (¬ (⟪M, N <fin N, M <fin ) → (((⟪M, N <fin N, M <fin ) M = N) → M = N))
3213, 30, 31sylc 56 1 (( Sfin (M, P) Sfin (N, P)) → M = N)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wo 357   wa 358   w3o 933   w3a 934  wex 1541   = wceq 1642   wcel 1710  wne 2516  c0 3550  cpw 3722  copk 4057  1cpw1 4135   Nn cnnc 4373   <fin cltfin 4433   Sfin wsfin 4438
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-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-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-pss 3261  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-fin 4380  df-lefin 4440  df-ltfin 4441  df-tfin 4443  df-sfin 4446
This theorem is referenced by:  vfinspss  4551
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