Theorem strsl0 13278

Description: All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)

Hypothesis

Ref	Expression
strsl0.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )

Assertion

Ref	Expression
strsl0	|- (/) = ( E ` (/) )

Proof of Theorem strsl0

Step	Hyp	Ref	Expression
1		0ex 4239	. . 3 |- (/) e. _V
2		strsl0.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpli 111	. . 3 |- E = Slot ( E ` ndx )
4	2	simpri 113	. . 3 |- ( E ` ndx ) e. NN
5	1, 3, 4	strnfvn 13250	. 2 |- ( E ` (/) ) = ( (/) ` ( E ` ndx ) )
6		0fv 5710	. 2 |- ( (/) ` ( E ` ndx ) ) = (/)
7	5, 6	eqtr2i 2256	1 |- (/) = ( E ` (/) )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   (/)c0 3510   ` cfv 5354   NNcn 9239   ndxcnx 13226  Slot cslot 13228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-slot 13233

This theorem is referenced by:  base0  13279  iedgval0  16066