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 2975	. 2 |- (Ord A -> (A i^i {A}) = (/))
2		disj3 2304	. . 3 |- ((A i^i {A}) = (/) <-> A = (A \ {A}))
3		df-suc 2944	. . . . . 6 |- suc A = (A u. {A})
4	3	difeq1i 2145	. . . . 5 |- (suc A \ {A}) = ((A u. {A}) \ {A})
5		difun2 2332	. . . . 5 |- ((A u. {A}) \ {A}) = (A \ {A})
6	4, 5	eqtr 1487	. . . 4 |- (suc A \ {A}) = (A \ {A})
7	6	eqeq2i 1477	. . 3 |- (A = (suc A \ {A}) <-> A = (A \ {A}))
8	2, 7	bitr4 176	. 2 |- ((A i^i {A}) = (/) <-> A = (suc A \ {A}))
9	1, 8	sylib 198	1 |- (Ord A -> A = (suc A \ {A}))

Syntax hints:   -> wi 3   = wceq 953   \ cdif 2034   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399  Ord word 2937  suc csuc 2940

This theorem is referenced by:  phplem3 4490  phplem4 4491  pssnn 4513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-eprel 2821  df-fr 2907  df-we 2924  df-ord 2941  df-suc 2944

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