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Theorem orddif 5962
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 5904 . 2 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2 disj3 4165 . . 3 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3 df-suc 5871 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
43difeq1i 3875 . . . . 5 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5 difun2 4191 . . . . 5 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
64, 5eqtri 2793 . . . 4 (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
76eqeq2i 2783 . . 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 1631  cdif 3720  cun 3721  cin 3722  c0 4063  {csn 4317  Ord word 5864  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163  df-fr 5209  df-we 5211  df-ord 5868  df-suc 5871
This theorem is referenced by:  phplem3  8301  phplem4  8302  pssnn  8338
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