Theorem orddif 2326

Description: Ordinal derived from its successor.

Assertion

Ref	Expression
orddif	⊢ (Ord A → A = (suc A ∖ {A}))

Proof of Theorem orddif

Step	Hyp	Ref	Expression
1		orddisj 2236	. 2 ⊢ (Ord A → (A ∩ {A}) = ∅)
2		disj3 1736	. . 3 ⊢ ((A ∩ {A}) = ∅ ↔ A = (A ∖ {A}))
3		df-suc 2205	. . . . . 6 ⊢ suc A = (A ∪ {A})
4	3	difeq1i 1584	. . . . 5 ⊢ (suc A ∖ {A}) = ((A ∪ {A}) ∖ {A})
5		difun2 1763	. . . . 5 ⊢ ((A ∪ {A}) ∖ {A}) = (A ∖ {A})
6	4, 5	eqtr 1119	. . . 4 ⊢ (suc A ∖ {A}) = (A ∖ {A})
7	6	cleq2i 1111	. . 3 ⊢ (A = (suc A ∖ {A}) ↔ A = (A ∖ {A}))
8	2, 7	bitr4 154	. 2 ⊢ ((A ∩ {A}) = ∅ ↔ A = (suc A ∖ {A}))
9	1, 8	sylib 173	1 ⊢ (Ord A → A = (suc A ∖ {A}))

Syntax hints:   → wi 2   = wceq 1091   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808  Ord word 2198  suc csuc 2201

This theorem is referenced by:  phplem4 3406  phplem5 3407  pssnn 3428

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202  df-suc 2205

metamath.org