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Theorem orddif 5737
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 5679 . 2 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2 disj3 3973 . . 3 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3 df-suc 5646 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
43difeq1i 3686 . . . . 5 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5 difun2 4000 . . . . 5 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
64, 5eqtri 2632 . . . 4 (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
76eqeq2i 2622 . . 3 (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴}))
82, 7bitr4i 266 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴}))
91, 8sylib 207 1 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  cdif 3537  cun 3538  cin 3539  c0 3874  {csn 4125  Ord word 5639  suc csuc 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997  df-we 4999  df-ord 5643  df-suc 5646
This theorem is referenced by:  phplem3  8026  phplem4  8027  pssnn  8063
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