Theorem sspw1or2 7544

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 2214	. . . . . 6 |- ( s = x -> ( j e. s <-> j e. x ) )
2	1	exbidv 1878	. . . . 5 |- ( s = x -> ( E. j j e. s <-> E. j j e. x ) )
3	2	elrab 2982	. . . 4 |- ( x e. { s e. ~P V | E. j j e. s } <-> ( x e. ~P V /\ E. j j e. x ) )
4	3	anbi1i 462	. . 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 7092	. . . . . 6 |- ( x ~~ 1o -> E. j j e. x )
6		en2m 7113	. . . . . 6 |- ( x ~~ 2o -> E. j j e. x )
7	5, 6	jaoi 728	. . . . 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 459	. . 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 2806	1 |- { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) }

This proof depends on syntax axioms:   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   ~Pcpw 3688   class class class wbr 4130   1oc1o 6680   2oc2o 6681   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1o 6687  df-2o 6688  df-en 7023

This theorem is used by:  subupgr  16514