Theorem orddif 3764

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 3701	. 2 |- (Ord A -> (A i^i {A}) = (/))
2		disj3 2918	. . 3 |- ((A i^i {A}) = (/) <-> A = (A \ {A}))
3		df-suc 3663	. . . . . 6 |- suc A = (A u. {A})
4	3	difeq1i 2722	. . . . 5 |- (suc A \ {A}) = ((A u. {A}) \ {A})
5		difun2 2953	. . . . 5 |- ((A u. {A}) \ {A}) = (A \ {A})
6	4, 5	eqtri 1908	. . . 4 |- (suc A \ {A}) = (A \ {A})
7	6	eqeq2i 1894	. . 3 |- (A = (suc A \ {A}) <-> A = (A \ {A}))
8	2, 7	bitr4i 193	. 2 |- ((A i^i {A}) = (/) <-> A = (suc A \ {A}))
9	1, 8	sylib 215	1 |- (Ord A -> A = (suc A \ {A}))

Syntax hints:   -> wi 3   = wceq 1298   \ cdif 2590   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044  Ord word 3656  suc csuc 3659

This theorem is referenced by:  phplem3 5604  phplem4 5605  pssnn 5628

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-fr 3625  df-we 3644  df-ord 3660  df-suc 3663

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