Theorem ufildom1 21730

Description: An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
ufildom1	|- ( F e. ( UFil ` X ) -> |^| F ~<_ 1o )

Proof of Theorem ufildom1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. 2 |- ( |^| F = (/) -> ( |^| F ~<_ 1o <-> (/) ~<_ 1o ) )
2		uffixsn 21729	. . . . . . . . 9 |- ( ( F e. ( UFil ` X ) /\ x e. |^| F ) -> { x } e. F )
3		intss1 4492	. . . . . . . . 9 |- ( { x } e. F -> |^| F C_ { x } )
4	2, 3	syl 17	. . . . . . . 8 |- ( ( F e. ( UFil ` X ) /\ x e. |^| F ) -> |^| F C_ { x } )
5		simpr 477	. . . . . . . . 9 |- ( ( F e. ( UFil ` X ) /\ x e. |^| F ) -> x e. |^| F )
6	5	snssd 4340	. . . . . . . 8 |- ( ( F e. ( UFil ` X ) /\ x e. |^| F ) -> { x } C_ |^| F )
7	4, 6	eqssd 3620	. . . . . . 7 |- ( ( F e. ( UFil ` X ) /\ x e. |^| F ) -> |^| F = { x } )
8	7	ex 450	. . . . . 6 |- ( F e. ( UFil ` X ) -> ( x e. |^| F -> |^| F = { x } ) )
9	8	eximdv 1846	. . . . 5 |- ( F e. ( UFil ` X ) -> ( E. x x e. |^| F -> E. x |^| F = { x } ) )
10		n0 3931	. . . . 5 |- ( |^| F =/= (/) <-> E. x x e. |^| F )
11		en1 8023	. . . . 5 |- ( |^| F ~~ 1o <-> E. x |^| F = { x } )
12	9, 10, 11	3imtr4g 285	. . . 4 |- ( F e. ( UFil ` X ) -> ( |^| F =/= (/) -> |^| F ~~ 1o ) )
13	12	imp 445	. . 3 |- ( ( F e. ( UFil ` X ) /\ |^| F =/= (/) ) -> |^| F ~~ 1o )
14		endom 7982	. . 3 |- ( |^| F ~~ 1o -> |^| F ~<_ 1o )
15	13, 14	syl 17	. 2 |- ( ( F e. ( UFil ` X ) /\ |^| F =/= (/) ) -> |^| F ~<_ 1o )
16		1on 7567	. . 3 |- 1o e. On
17		0domg 8087	. . 3 |- ( 1o e. On -> (/) ~<_ 1o )
18	16, 17	mp1i 13	. 2 |- ( F e. ( UFil ` X ) -> (/) ~<_ 1o )
19	1, 15, 18	pm2.61ne 2879	1 |- ( F e. ( UFil ` X ) -> |^| F ~<_ 1o )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   {csn 4177   |^|cint 4475   class class class wbr 4653   Oncon0 5723   ` cfv 5888   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   UFilcufil 21703

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-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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-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-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-ord 5726  df-on 5727  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-en 7956  df-dom 7957  df-fbas 19743  df-fg 19744  df-fil 21650  df-ufil 21705

This theorem is referenced by: (None)