Theorem orddif 3075

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 2985	. 2 |- (Ord A -> (A i^i {A}) = (/))
2		disj3 2314	. . 3 |- ((A i^i {A}) = (/) <-> A = (A \ {A}))
3		df-suc 2954	. . . . . 6 |- suc A = (A u. {A})
4	3	difeq1i 2155	. . . . 5 |- (suc A \ {A}) = ((A u. {A}) \ {A})
5		difun2 2342	. . . . 5 |- ((A u. {A}) \ {A}) = (A \ {A})
6	4, 5	eqtr 1495	. . . 4 |- (suc A \ {A}) = (A \ {A})
7	6	eqeq2i 1485	. . 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 956   \ cdif 2044   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409  Ord word 2947  suc csuc 2950

This theorem is referenced by:  phplem3 4510  phplem4 4511  pssnn 4534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-fr 2917  df-we 2934  df-ord 2951  df-suc 2954

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