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Theorem orddif 4299
 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 4298 . 2 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2 disj3 3300 . . 3 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3 df-suc 4136 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
43difeq1i 3086 . . . . 5 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5 difun2 3330 . . . . 5 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
64, 5eqtri 2076 . . . 4 (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
76eqeq2i 2066 . . 3 (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴}))
82, 7bitr4i 180 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴}))
91, 8sylib 131 1 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∖ cdif 2942   ∪ cun 2943   ∩ cin 2944  ∅c0 3252  {csn 3403  Ord word 4127  suc csuc 4130 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-suc 4136 This theorem is referenced by:  phplem3  6348  phplem4  6349  phplem4dom  6355  phplem4on  6360  dif1en  6368
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