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Theorem orddif 5858
 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 5800 . 2 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2 disj3 4054 . . 3 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3 df-suc 5767 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
43difeq1i 3757 . . . . 5 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5 difun2 4081 . . . . 5 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
64, 5eqtri 2673 . . . 4 (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
76eqeq2i 2663 . . 3 (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴}))
82, 7bitr4i 267 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴}))
91, 8sylib 208 1 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  Ord word 5760  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-fr 5102  df-we 5104  df-ord 5764  df-suc 5767 This theorem is referenced by:  phplem3  8182  phplem4  8183  pssnn  8219
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