Theorem orddif 3065

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 3013	. 2 |- (Ord A -> (A i^i {A}) = (/))
2		disj3 2367	. . 3 |- ((A i^i {A}) = (/) <-> A = (A \ {A}))
3		df-suc 2981	. . . . . 6 |- suc A = (A u. {A})
4	3	difeq1i 2207	. . . . 5 |- (suc A \ {A}) = ((A u. {A}) \ {A})
5		difun2 2396	. . . . 5 |- ((A u. {A}) \ {A}) = (A \ {A})
6	4, 5	eqtri 1538	. . . 4 |- (suc A \ {A}) = (A \ {A})
7	6	eqeq2i 1528	. . 3 |- (A = (suc A \ {A}) <-> A = (A \ {A}))
8	2, 7	bitr4i 174	. 2 |- ((A i^i {A}) = (/) <-> A = (suc A \ {A}))
9	1, 8	sylib 196	1 |- (Ord A -> A = (suc A \ {A}))

Syntax hints:   -> wi 3   = wceq 992   \ cdif 2096   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467  Ord word 2974  suc csuc 2977

This theorem is referenced by:  phplem3 4657  phplem4 4658  pssnn 4681

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693  df-opab 2741  df-eprel 2910  df-fr 2947  df-we 2962  df-ord 2978  df-suc 2981

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