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Theorem orddif 6281
Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
orddif (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Proof of Theorem orddif
StepHypRef Expression
1 orddisj 6226 . 2 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2 disj3 4405 . . 3 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3 df-suc 6194 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
43difeq1i 4098 . . . . 5 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5 difun2 4431 . . . . 5 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
64, 5eqtri 2848 . . . 4 (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
76eqeq2i 2838 . . 3 (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴}))
82, 7bitr4i 279 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴}))
91, 8sylib 219 1 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  cdif 3936  cun 3937  cin 3938  c0 4294  {csn 4563  Ord word 6187  suc csuc 6190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-eprel 5463  df-fr 5512  df-we 5514  df-ord 6191  df-suc 6194
This theorem is referenced by:  phplem3  8690  phplem4  8691  pssnn  8728
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