Theorem pw1ndom3 16610

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 4289). (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 16608	. . 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 544	. . . . . . . . . 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 536	. . . . . . . . . 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 529	. . . . . . . . . 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 1029	. . . . . . . . . 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 1030	. . . . . . . . . 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 1031	. . . . . . . . . 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 16609	. . . . . . . . 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 2488	. . . . . . . . . 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 16609	. . . . . . . . 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 2267	. . . . . . . 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 2485	. . . . . . 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 2650	. . . . 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 2650	. . . 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 2650	. . 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 1415	. 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 1004   F. wfal 1402   e. wcel 2202   =/= wne 2402   E.wrex 2511   (/)c0 3494   ~Pcpw 3652   class class class wbr 4088   1oc1o 6575   3oc3o 6577   ~<_ cdom 6908

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  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  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-ne 2403  df-ral 2515  df-rex 2516  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-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-1o 6582  df-2o 6583  df-3o 6584  df-dom 6911

This theorem is referenced by:  pw1ninf  16611