Theorem pw1ndom3 16383

Description: The powerset of 1o does not dominate 3o. This is another way of saying that ~P 1o does not have three elements (like pwntru 4283). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ndom3	|- -. 3o ~<_ ~P 1o

Proof of Theorem pw1ndom3

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3dom 16381	. . 3 |- ( 3o ~<_ ~P 1o -> E. x e. ~P 1o E. y e. ~P 1o E. z e. ~P 1o ( x =/= y /\ x =/= z /\ y =/= z ) )
2		simp-4r 542	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x e. ~P 1o )
3		simpllr 534	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y e. ~P 1o )
4		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> z e. ~P 1o )
5		simpr1 1027	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x =/= y )
6		simpr2 1028	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x =/= z )
7		simpr3 1029	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y =/= z )
8	2, 3, 4, 5, 6, 7	pw1ndom3lem 16382	. . . . . . . . 9 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x = (/) )
9	5	necomd 2486	. . . . . . . . . 10 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y =/= x )
10	3, 2, 4, 9, 7, 6	pw1ndom3lem 16382	. . . . . . . . 9 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y = (/) )
11	8, 10	eqtr4d 2265	. . . . . . . 8 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x = y )
12	11, 5	pm2.21ddne 2483	. . . . . . 7 |- ( ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> F. )
13	12	ex 115	. . . . . 6 |- ( ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) /\ z e. ~P 1o ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) -> F. ) )
14	13	rexlimdva 2648	. . . . 5 |- ( ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) /\ y e. ~P 1o ) -> ( E. z e. ~P 1o ( x =/= y /\ x =/= z /\ y =/= z ) -> F. ) )
15	14	rexlimdva 2648	. . . 4 |- ( ( 3o ~<_ ~P 1o /\ x e. ~P 1o ) -> ( E. y e. ~P 1o E. z e. ~P 1o ( x =/= y /\ x =/= z /\ y =/= z ) -> F. ) )
16	15	rexlimdva 2648	. . 3 |- ( 3o ~<_ ~P 1o -> ( E. x e. ~P 1o E. y e. ~P 1o E. z e. ~P 1o ( x =/= y /\ x =/= z /\ y =/= z ) -> F. ) )
17	1, 16	mpd 13	. 2 |- ( 3o ~<_ ~P 1o -> F. )
18		dfnot 1413	. 2 |- ( -. 3o ~<_ ~P 1o <-> ( 3o ~<_ ~P 1o -> F. ) )
19	17, 18	mpbir 146	1 |- -. 3o ~<_ ~P 1o

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1002   F. wfal 1400   e. wcel 2200   =/= wne 2400   E.wrex 2509   (/)c0 3491   ~Pcpw 3649   class class class wbr 4083   1oc1o 6561   3oc3o 6563   ~<_ cdom 6894

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fv 5326  df-1o 6568  df-2o 6569  df-3o 6570  df-dom 6897

This theorem is referenced by:  pw1ninf  16384