Theorem sfin111 4537

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.

Step	Hyp	Ref	Expression
1		df-sfin 4447	. . . . . . 7 ⊢ ( Sfin (N, P) ↔ (N ∈ Nn ∧ P ∈ Nn ∧ ∃a(℘1a ∈ N ∧ ℘a ∈ P)))
2	1	simp2bi 971	. . . . . 6 ⊢ ( Sfin (N, P) → P ∈ Nn )
3	2	adantl 452	. . . . 5 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → P ∈ Nn )
4		ltfinirr 4458	. . . . 5 ⊢ (P ∈ Nn → ¬ ⟪P, P⟫ ∈ <fin )
5	3, 4	syl 15	. . . 4 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → ¬ ⟪P, P⟫ ∈ <fin )
6		sfinltfin 4536	. . . 4 ⊢ ((( Sfin (M, P) ∧ Sfin (N, P)) ∧ ⟪M, N⟫ ∈ <fin ) → ⟪P, P⟫ ∈ <fin )
7	5, 6	mtand 640	. . 3 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → ¬ ⟪M, N⟫ ∈ <fin )
8		sfinltfin 4536	. . . . . 6 ⊢ ((( Sfin (N, P) ∧ Sfin (M, P)) ∧ ⟪N, M⟫ ∈ <fin ) → ⟪P, P⟫ ∈ <fin )
9	8	ex 423	. . . . 5 ⊢ (( Sfin (N, P) ∧ Sfin (M, P)) → (⟪N, M⟫ ∈ <fin → ⟪P, P⟫ ∈ <fin ))
10	9	ancoms 439	. . . 4 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → (⟪N, M⟫ ∈ <fin → ⟪P, P⟫ ∈ <fin ))
11	5, 10	mtod 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 ))
13	7, 11, 12	sylanbrc 645	. 2 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → ¬ (⟪M, N⟫ ∈ <fin ∨ ⟪N, M⟫ ∈ <fin ))
14		df-sfin 4447	. . . . . . 7 ⊢ ( Sfin (M, P) ↔ (M ∈ Nn ∧ P ∈ Nn ∧ ∃a(℘1a ∈ M ∧ ℘a ∈ P)))
15	14	simp1bi 970	. . . . . 6 ⊢ ( Sfin (M, P) → M ∈ Nn )
16	15	adantr 451	. . . . 5 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → M ∈ Nn )
17	1	simp1bi 970	. . . . . 6 ⊢ ( Sfin (N, P) → N ∈ Nn )
18	17	adantl 452	. . . . 5 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → N ∈ Nn )
19		ne0i 3557	. . . . . . . . . 10 ⊢ (℘1a ∈ M → M ≠ ∅)
20	19	adantr 451	. . . . . . . . 9 ⊢ ((℘1a ∈ M ∧ ℘a ∈ P) → M ≠ ∅)
21	20	exlimiv 1634	. . . . . . . 8 ⊢ (∃a(℘1a ∈ M ∧ ℘a ∈ P) → M ≠ ∅)
22	21	3ad2ant3 978	. . . . . . 7 ⊢ ((M ∈ Nn ∧ P ∈ Nn ∧ ∃a(℘1a ∈ M ∧ ℘a ∈ P)) → M ≠ ∅)
23	14, 22	sylbi 187	. . . . . 6 ⊢ ( Sfin (M, P) → M ≠ ∅)
24	23	adantr 451	. . . . 5 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → M ≠ ∅)
25		ltfintri 4467	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ M ≠ ∅) → (⟪M, N⟫ ∈ <fin ∨ M = N ∨ ⟪N, M⟫ ∈ <fin ))
26	16, 18, 24, 25	syl3anc 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 ))
28	26, 27	sylib 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))
30	28, 29	sylib 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))
32	13, 30, 31	sylc 56	1 ⊢ (( Sfin (M, P) ∧ Sfin (N, P)) → M = N)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551  ℘cpw 3723  ⟪copk 4058  ℘₁cpw1 4136   Nn cnnc 4374   <_fin cltfin 4434   S_fin wsfin 4439

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-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-pss 3262  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-fin 4381  df-lefin 4441  df-ltfin 4442  df-tfin 4444  df-sfin 4447

This theorem is used by:  vfinspss  4552