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Theorem 0cminle 4461
 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, Afin )

Proof of Theorem 0cminle
StepHypRef Expression
1 addcid2 4407 . . 3 (0c +c A) = A
21opkeq2i 4063 . 2 ⟪0c, (0c +c A)⟫ = ⟪0c, A
3 peano1 4402 . . 3 0c Nn
4 lefinaddc 4450 . . 3 ((0c Nn A Nn ) → ⟪0c, (0c +c A)⟫ fin )
53, 4mpan 651 . 2 (A Nn → ⟪0c, (0c +c A)⟫ fin )
62, 5syl5eqelr 2438 1 (A Nn → ⟪0c, Afin )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ⟪copk 4057   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   ≤fin clefin 4432 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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-pw 3724  df-sn 3741  df-pr 3742  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379  df-lefin 4440 This theorem is referenced by:  ltfintri  4466
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