Theorem pw0ss 15963

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 3821	. . 3 |- ~P (/) = { (/) }
2	1	rabeqi 2794	. 2 |- { s e. ~P (/) | E. j j e. s } = { s e. { (/) } | E. j j e. s }
3		rabeq0 3523	. . 3 |- ( { s e. { (/) } | E. j j e. s } = (/) <-> A. s e. { (/) } -. E. j j e. s )
4		elsni 3688	. . . 4 |- ( s e. { (/) } -> s = (/) )
5		notm0 3514	. . . 4 |- ( -. E. j j e. s <-> s = (/) )
6	4, 5	sylibr 134	. . 3 |- ( s e. { (/) } -> -. E. j j e. s )
7	3, 6	mprgbir 2589	. 2 |- { s e. { (/) } | E. j j e. s } = (/)
8	2, 7	eqtri 2251	1 |- { s e. ~P (/) | E. j j e. s } = (/)

Syntax hints:   -. wn 3   = wceq 1397   E.wex 1540   e. wcel 2201   {crab 2513   (/)c0 3493   ~Pcpw 3653   {csn 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rab 2518  df-v 2803  df-dif 3201  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676

This theorem is referenced by:  uhgr0vb  15964  uhgr0  15965