Theorem n0lpligALT 27336

Description: Alternate version of n0lplig 27335 using the predicate e/ instead of -. e. and whose proof bypasses nsnlplig 27333. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
n0lpligALT	|- ( G e. Plig -> (/) e/ G )

Proof of Theorem n0lpligALT

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- U. G = U. G
2	1	l2p 27331	. . 3 |- ( ( G e. Plig /\ (/) e. G ) -> E. a e. U. G E. b e. U. G ( a =/= b /\ a e. (/) /\ b e. (/) ) )
3		noel 3919	. . . . . . 7 |- -. a e. (/)
4	3	pm2.21i 116	. . . . . 6 |- ( a e. (/) -> (/) e/ G )
5	4	3ad2ant2 1083	. . . . 5 |- ( ( a =/= b /\ a e. (/) /\ b e. (/) ) -> (/) e/ G )
6	5	a1i 11	. . . 4 |- ( ( a e. U. G /\ b e. U. G ) -> ( ( a =/= b /\ a e. (/) /\ b e. (/) ) -> (/) e/ G ) )
7	6	rexlimivv 3036	. . 3 |- ( E. a e. U. G E. b e. U. G ( a =/= b /\ a e. (/) /\ b e. (/) ) -> (/) e/ G )
8	2, 7	syl 17	. 2 |- ( ( G e. Plig /\ (/) e. G ) -> (/) e/ G )
9		simpr 477	. 2 |- ( ( G e. Plig /\ (/) e/ G ) -> (/) e/ G )
10	8, 9	pm2.61danel 2911	1 |- ( G e. Plig -> (/) e/ G )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   (/)c0 3915   U.cuni 4436   Pligcplig 27326

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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-v 3202  df-dif 3577  df-nul 3916  df-uni 4437  df-plig 27327

This theorem is referenced by: (None)