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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 A V) → ⟪A, fin )

Proof of Theorem nulge
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 addcnul1 4452 . . . . 5 (A +c ) =
21eqcomi 2357 . . . 4 = (A +c )
3 addceq2 4384 . . . . . 6 (x = → (A +c x) = (A +c ))
43eqeq2d 2364 . . . . 5 (x = → ( = (A +c x) ↔ = (A +c )))
54rspcev 2955 . . . 4 (( Nn = (A +c )) → x Nn = (A +c x))
62, 5mpan2 652 . . 3 ( Nnx Nn = (A +c x))
76adantr 451 . 2 (( Nn A V) → x Nn = (A +c x))
8 opklefing 4448 . . 3 ((A V Nn ) → (⟪A, finx Nn = (A +c x)))
98ancoms 439 . 2 (( Nn A V) → (⟪A, finx Nn = (A +c x)))
107, 9mpbird 223 1 (( Nn A V) → ⟪A, 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   +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-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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