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Theorem nulge 4456
 Description: If the empty set is a finite cardinal, then it is a maximal element. (Contributed by SF, 19-Jan-2015.)
Assertion
Ref Expression
nulge Nn fin

Proof of Theorem nulge
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 addcnul1 4452 . . . . 5
21eqcomi 2357 . . . 4
3 addceq2 4384 . . . . . 6
43eqeq2d 2364 . . . . 5
54rspcev 2955 . . . 4 Nn Nn
62, 5mpan2 652 . . 3 Nn Nn
76adantr 451 . 2 Nn Nn
8 opklefing 4448 . . 3 Nn fin Nn
98ancoms 439 . 2 Nn fin Nn
107, 9mpbird 223 1 Nn fin
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wrex 2615  c0 3550  copk 4057   Nn cnnc 4373   cplc 4375   fin clefin 4432 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 4078  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-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-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-addc 4378  df-lefin 4440 This theorem is referenced by:  lenltfin  4469
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