Theorem orduniorsuc 7030

Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)

Assertion

Ref	Expression
orduniorsuc	|- ( Ord A -> ( A = U. A \/ A = suc U. A ) )

Proof of Theorem orduniorsuc

Step	Hyp	Ref	Expression
1		orduniss 5821	. . . . . 6 |- ( Ord A -> U. A C_ A )
2		orduni 6994	. . . . . . . 8 |- ( Ord A -> Ord U. A )
3		ordelssne 5750	. . . . . . . 8 |- ( ( Ord U. A /\ Ord A ) -> ( U. A e. A <-> ( U. A C_ A /\ U. A =/= A ) ) )
4	2, 3	mpancom 703	. . . . . . 7 |- ( Ord A -> ( U. A e. A <-> ( U. A C_ A /\ U. A =/= A ) ) )
5	4	biimprd 238	. . . . . 6 |- ( Ord A -> ( ( U. A C_ A /\ U. A =/= A ) -> U. A e. A ) )
6	1, 5	mpand 711	. . . . 5 |- ( Ord A -> ( U. A =/= A -> U. A e. A ) )
7		ordsucss 7018	. . . . 5 |- ( Ord A -> ( U. A e. A -> suc U. A C_ A ) )
8	6, 7	syld 47	. . . 4 |- ( Ord A -> ( U. A =/= A -> suc U. A C_ A ) )
9		ordsucuni 7029	. . . 4 |- ( Ord A -> A C_ suc U. A )
10	8, 9	jctild 566	. . 3 |- ( Ord A -> ( U. A =/= A -> ( A C_ suc U. A /\ suc U. A C_ A ) ) )
11		df-ne 2795	. . . 4 |- ( A =/= U. A <-> -. A = U. A )
12		necom 2847	. . . 4 |- ( A =/= U. A <-> U. A =/= A )
13	11, 12	bitr3i 266	. . 3 |- ( -. A = U. A <-> U. A =/= A )
14		eqss 3618	. . 3 |- ( A = suc U. A <-> ( A C_ suc U. A /\ suc U. A C_ A ) )
15	10, 13, 14	3imtr4g 285	. 2 |- ( Ord A -> ( -. A = U. A -> A = suc U. A ) )
16	15	orrd 393	1 |- ( Ord A -> ( A = U. A \/ A = suc U. A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   U.cuni 4436   Ord word 5722   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  onuniorsuci  7039  oeeulem  7681  cantnfp1lem2  8576  cantnflem1  8586  cnfcom2lem  8598  dfac12lem1  8965  dfac12lem2  8966  ttukeylem3  9333  ttukeylem5  9335  ttukeylem6  9336  ordtoplem  32434  ordcmp  32446  onsucuni3  33215  aomclem5  37628  onsetreclem3  42450