Theorem bj-1upln0 32997

Description: A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-1upln0	|- (| A|) =/= (/)

Proof of Theorem bj-1upln0

Step	Hyp	Ref	Expression
1		df-bj-1upl 32986	. 2 |- (| A|) = ( { (/) } X. tag A )
2		0nep0 4836	. . . 4 |- (/) =/= { (/) }
3	2	necomi 2848	. . 3 |- { (/) } =/= (/)
4		bj-tagn0 32967	. . 3 |- tag A =/= (/)
5		xpnz 5553	. . . 4 |- ( ( { (/) } =/= (/) /\ tag A =/= (/) ) <-> ( { (/) } X. tag A ) =/= (/) )
6	5	biimpi 206	. . 3 |- ( ( { (/) } =/= (/) /\ tag A =/= (/) ) -> ( { (/) } X. tag A ) =/= (/) )
7	3, 4, 6	mp2an 708	. 2 |- ( { (/) } X. tag A ) =/= (/)
8	1, 7	eqnetri 2864	1 |- (| A|) =/= (/)

Syntax hints:   /\ wa 384   =/= wne 2794   (/)c0 3915   {csn 4177   X. cxp 5112  tag bj-ctag 32962  (|bj-c1upl 32985

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-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-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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-bj-tag 32963  df-bj-1upl 32986

This theorem is referenced by:  bj-2upln0  33011