Theorem elwlim 31769

Description: Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)

Assertion

Ref	Expression
elwlim	|- ( X e. WLim ( R , A ) <-> ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) )

Proof of Theorem elwlim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2856	. . . 4 |- ( x = X -> ( x =/= inf ( A , A , R ) <-> X =/= inf ( A , A , R ) ) )
2		id 22	. . . . 5 |- ( x = X -> x = X )
3		predeq3 5684	. . . . . 6 |- ( x = X -> Pred ( R , A , x ) = Pred ( R , A , X ) )
4	3	supeq1d 8352	. . . . 5 |- ( x = X -> sup ( Pred ( R , A , x ) , A , R ) = sup ( Pred ( R , A , X ) , A , R ) )
5	2, 4	eqeq12d 2637	. . . 4 |- ( x = X -> ( x = sup ( Pred ( R , A , x ) , A , R ) <-> X = sup ( Pred ( R , A , X ) , A , R ) ) )
6	1, 5	anbi12d 747	. . 3 |- ( x = X -> ( ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) <-> ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
7		df-wlim 31758	. . 3 |- WLim ( R , A ) = { x e. A | ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) }
8	6, 7	elrab2 3366	. 2 |- ( X e. WLim ( R , A ) <-> ( X e. A /\ ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
9		3anass 1042	. 2 |- ( ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) <-> ( X e. A /\ ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
10	8, 9	bitr4i 267	1 |- ( X e. WLim ( R , A ) <-> ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   Predcpred 5679   supcsup 8346  infcinf 8347  WLimcwlim 31754

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

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-ne 2795  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-sup 8348  df-wlim 31758

This theorem is referenced by: (None)