Theorem 0cminle 4462

Description: Cardinal zero is a minimal element for finite less than or equal. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
0cminle	⊢ (A ∈ Nn → ⟪0c, A⟫ ∈ ≤fin )

Proof of Theorem 0cminle

Step	Hyp	Ref	Expression
1		addcid2 4408	. . 3 ⊢ (0c +c A) = A
2	1	opkeq2i 4064	. 2 ⊢ ⟪0c, (0c +c A)⟫ = ⟪0c, A⟫
3		peano1 4403	. . 3 ⊢ 0c ∈ Nn
4		lefinaddc 4451	. . 3 ⊢ ((0c ∈ Nn ∧ A ∈ Nn ) → ⟪0c, (0c +c A)⟫ ∈ ≤fin )
5	3, 4	mpan 651	. 2 ⊢ (A ∈ Nn → ⟪0c, (0c +c A)⟫ ∈ ≤fin )
6	2, 5	syl5eqelr 2438	1 ⊢ (A ∈ Nn → ⟪0c, A⟫ ∈ ≤fin )

Syntax hints:   → wi 4   ∈ wcel 1710  ⟪copk 4058   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   ≤_fin clefin 4433

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-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 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-pw 3725  df-sn 3742  df-pr 3743  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379  df-nnc 4380  df-lefin 4441

This theorem is referenced by:  ltfintri  4467