Theorem psubatN 35041

Description: A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
atpsub.a	|- A = ( Atoms ` K )
atpsub.s	|- S = ( PSubSp ` K )

Assertion

Ref	Expression
psubatN	|- ( ( K e. B /\ X e. S /\ Y e. X ) -> Y e. A )

Proof of Theorem psubatN

Step	Hyp	Ref	Expression
1		atpsub.a	. . . 4 |- A = ( Atoms ` K )
2		atpsub.s	. . . 4 |- S = ( PSubSp ` K )
3	1, 2	psubssat 35040	. . 3 |- ( ( K e. B /\ X e. S ) -> X C_ A )
4	3	sseld 3602	. 2 |- ( ( K e. B /\ X e. S ) -> ( Y e. X -> Y e. A ) )
5	4	3impia 1261	1 |- ( ( K e. B /\ X e. S /\ Y e. X ) -> Y e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888   Atomscatm 34550   PSubSpcpsubsp 34782

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-psubsp 34789

This theorem is referenced by: (None)