Theorem pw0ss 16127

Description: There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)

Assertion

Ref	Expression
pw0ss	|- { s e. ~P (/) | E. j j e. s } = (/)

Distinct variable group:   j, s

Proof of Theorem pw0ss

Step	Hyp	Ref	Expression
1		pw0 3843	. . 3 |- ~P (/) = { (/) }
2	1	rabeqi 2808	. 2 |- { s e. ~P (/) | E. j j e. s } = { s e. { (/) } | E. j j e. s }
3		rabeq0 3540	. . 3 |- ( { s e. { (/) } | E. j j e. s } = (/) <-> A. s e. { (/) } -. E. j j e. s )
4		elsni 3709	. . . 4 |- ( s e. { (/) } -> s = (/) )
5		notm0 3531	. . . 4 |- ( -. E. j j e. s <-> s = (/) )
6	4, 5	sylibr 134	. . 3 |- ( s e. { (/) } -> -. E. j j e. s )
7	3, 6	mprgbir 2602	. 2 |- { s e. { (/) } | E. j j e. s } = (/)
8	2, 7	eqtri 2255	1 |- { s e. ~P (/) | E. j j e. s } = (/)

Syntax hints:   -. wn 3   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   (/)c0 3510   ~Pcpw 3671   {csn 3691

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-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rab 2531  df-v 2817  df-dif 3215  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697

This theorem is referenced by:  uhgr0vb  16128  uhgr0  16129