Theorem sspw1or2 7402

Description: The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)

Assertion

Ref	Expression
sspw1or2	|- { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) }

Distinct variable groups:   V, s   j, s, x

Allowed substitution hints:   V( x, j)

Proof of Theorem sspw1or2

Step	Hyp	Ref	Expression
1		elequ2 2207	. . . . . 6 |- ( s = x -> ( j e. s <-> j e. x ) )
2	1	exbidv 1873	. . . . 5 |- ( s = x -> ( E. j j e. s <-> E. j j e. x ) )
3	2	elrab 2962	. . . 4 |- ( x e. { s e. ~P V | E. j j e. s } <-> ( x e. ~P V /\ E. j j e. x ) )
4	3	anbi1i 458	. . 3 |- ( ( x e. { s e. ~P V | E. j j e. s } /\ ( x ~~ 1o \/ x ~~ 2o ) ) <-> ( ( x e. ~P V /\ E. j j e. x ) /\ ( x ~~ 1o \/ x ~~ 2o ) ) )
5		en1m 6978	. . . . . 6 |- ( x ~~ 1o -> E. j j e. x )
6		en2m 6998	. . . . . 6 |- ( x ~~ 2o -> E. j j e. x )
7	5, 6	jaoi 723	. . . . 5 |- ( ( x ~~ 1o \/ x ~~ 2o ) -> E. j j e. x )
8	7	biantrud 304	. . . 4 |- ( ( x ~~ 1o \/ x ~~ 2o ) -> ( x e. ~P V <-> ( x e. ~P V /\ E. j j e. x ) ) )
9	8	pm5.32ri 455	. . 3 |- ( ( x e. ~P V /\ ( x ~~ 1o \/ x ~~ 2o ) ) <-> ( ( x e. ~P V /\ E. j j e. x ) /\ ( x ~~ 1o \/ x ~~ 2o ) ) )
10	4, 9	bitr4i 187	. 2 |- ( ( x e. { s e. ~P V | E. j j e. s } /\ ( x ~~ 1o \/ x ~~ 2o ) ) <-> ( x e. ~P V /\ ( x ~~ 1o \/ x ~~ 2o ) ) )
11	10	rabbia2 2787	1 |- { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) }

Syntax hints:   /\ wa 104   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   {crab 2514   ~Pcpw 3652   class class class wbr 4088   1oc1o 6574   2oc2o 6575   ~~ cen 6906

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-en 6909

This theorem is referenced by:  subupgr  16123