Theorem ldlfcntref 29921

Description: Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)

Hypothesis

Ref	Expression
ldlfcntref.x	|- X = U. J

Assertion

Ref	Expression
ldlfcntref	|- ( ( J e. Ldlf /\ U C_ J /\ X = U. U ) -> E. v e. ~P J ( v ~<_ om /\ v Ref U ) )

Distinct variable groups:   v, J   v, U

Allowed substitution hint:   X( v)

Proof of Theorem ldlfcntref

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldlfcntref.x	. 2 |- X = U. J
2		df-ldlf 29920	. 2 |- Ldlf = CovHasRef { x | x ~<_ om }
3		vex 3203	. . . 4 |- v e. _V
4		breq1 4656	. . . 4 |- ( x = v -> ( x ~<_ om <-> v ~<_ om ) )
5	3, 4	elab 3350	. . 3 |- ( v e. { x | x ~<_ om } <-> v ~<_ om )
6	5	biimpi 206	. 2 |- ( v e. { x | x ~<_ om } -> v ~<_ om )
7	1, 2, 6	crefdf 29915	1 |- ( ( J e. Ldlf /\ U C_ J /\ X = U. U ) -> E. v e. ~P J ( v ~<_ om /\ v Ref U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   omcom 7065   ~<_ cdom 7953   Refcref 21305  Ldlfcldlf 29919

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-cref 29910  df-ldlf 29920

This theorem is referenced by: (None)