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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
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