Theorem cfslbn 9089

Description: Any subset of A smaller than its cofinality has union less than A. (This is the contrapositive to cfslb 9088.) (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfslb.1	|- A e. _V

Assertion

Ref	Expression
cfslbn	|- ( ( Lim A /\ B C_ A /\ B ~< ( cf ` A ) ) -> U. B e. A )

Proof of Theorem cfslbn

Step	Hyp	Ref	Expression
1		uniss 4458	. . . . . . . 8 |- ( B C_ A -> U. B C_ U. A )
2		limuni 5785	. . . . . . . . 9 |- ( Lim A -> A = U. A )
3	2	sseq2d 3633	. . . . . . . 8 |- ( Lim A -> ( U. B C_ A <-> U. B C_ U. A ) )
4	1, 3	syl5ibr 236	. . . . . . 7 |- ( Lim A -> ( B C_ A -> U. B C_ A ) )
5	4	imp 445	. . . . . 6 |- ( ( Lim A /\ B C_ A ) -> U. B C_ A )
6		limord 5784	. . . . . . . . . . . 12 |- ( Lim A -> Ord A )
7		ordsson 6989	. . . . . . . . . . . 12 |- ( Ord A -> A C_ On )
8	6, 7	syl 17	. . . . . . . . . . 11 |- ( Lim A -> A C_ On )
9		sstr2 3610	. . . . . . . . . . 11 |- ( B C_ A -> ( A C_ On -> B C_ On ) )
10	8, 9	syl5com 31	. . . . . . . . . 10 |- ( Lim A -> ( B C_ A -> B C_ On ) )
11		ssorduni 6985	. . . . . . . . . 10 |- ( B C_ On -> Ord U. B )
12	10, 11	syl6 35	. . . . . . . . 9 |- ( Lim A -> ( B C_ A -> Ord U. B ) )
13	12, 6	jctird 567	. . . . . . . 8 |- ( Lim A -> ( B C_ A -> ( Ord U. B /\ Ord A ) ) )
14		ordsseleq 5752	. . . . . . . 8 |- ( ( Ord U. B /\ Ord A ) -> ( U. B C_ A <-> ( U. B e. A \/ U. B = A ) ) )
15	13, 14	syl6 35	. . . . . . 7 |- ( Lim A -> ( B C_ A -> ( U. B C_ A <-> ( U. B e. A \/ U. B = A ) ) ) )
16	15	imp 445	. . . . . 6 |- ( ( Lim A /\ B C_ A ) -> ( U. B C_ A <-> ( U. B e. A \/ U. B = A ) ) )
17	5, 16	mpbid 222	. . . . 5 |- ( ( Lim A /\ B C_ A ) -> ( U. B e. A \/ U. B = A ) )
18	17	ord 392	. . . 4 |- ( ( Lim A /\ B C_ A ) -> ( -. U. B e. A -> U. B = A ) )
19		cfslb.1	. . . . . . 7 |- A e. _V
20	19	cfslb 9088	. . . . . 6 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) ~<_ B )
21		domnsym 8086	. . . . . 6 |- ( ( cf ` A ) ~<_ B -> -. B ~< ( cf ` A ) )
22	20, 21	syl 17	. . . . 5 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> -. B ~< ( cf ` A ) )
23	22	3expia 1267	. . . 4 |- ( ( Lim A /\ B C_ A ) -> ( U. B = A -> -. B ~< ( cf ` A ) ) )
24	18, 23	syld 47	. . 3 |- ( ( Lim A /\ B C_ A ) -> ( -. U. B e. A -> -. B ~< ( cf ` A ) ) )
25	24	con4d 114	. 2 |- ( ( Lim A /\ B C_ A ) -> ( B ~< ( cf ` A ) -> U. B e. A ) )
26	25	3impia 1261	1 |- ( ( Lim A /\ B C_ A /\ B ~< ( cf ` A ) ) -> U. B e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   Ord word 5722   Oncon0 5723   Lim wlim 5724   ` cfv 5888   ~<_ cdom 7953   ~< csdm 7954   cfccf 8763

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-cf 8767

This theorem is referenced by:  cfslb2n  9090