Theorem pw1ndom3 16764

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 4312). (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 16762	. . 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 1030	. . . . . . . . . 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 1031	. . . . . . . . . 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 1032	. . . . . . . . . 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 16763	. . . . . . . . 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 2498	. . . . . . . . . 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 16763	. . . . . . . . 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 2268	. . . . . . . 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 2495	. . . . . . 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 2660	. . . . 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 2660	. . . 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 2660	. . 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 1416	. 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 1005   F. wfal 1403   e. wcel 2203   =/= wne 2412   E.wrex 2521   (/)c0 3508   ~Pcpw 3669   class class class wbr 4109   1oc1o 6640   3oc3o 6642   ~<_ cdom 6974

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fv 5360  df-1o 6647  df-2o 6648  df-3o 6649  df-dom 6977

This theorem is referenced by:  pw1ninf  16765