Theorem pw1ndom3 16934

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 4331). (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 16932	. . 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 548	. . . . . . . . . 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 540	. . . . . . . . . 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 533	. . . . . . . . . 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 1034	. . . . . . . . . 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 1035	. . . . . . . . . 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 1036	. . . . . . . . . 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 16933	. . . . . . . . 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 2506	. . . . . . . . . 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 16933	. . . . . . . . 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 2274	. . . . . . . 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 2503	. . . . . . 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 2668	. . . . 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 2668	. . . 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 2668	. . 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 1420	. 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 1009   F. wfal 1407   e. wcel 2209   =/= wne 2420   E.wrex 2529   (/)c0 3520   ~Pcpw 3685   class class class wbr 4125   1oc1o 6670   3oc3o 6672   ~<_ cdom 7011

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fv 5380  df-1o 6677  df-2o 6678  df-3o 6679  df-dom 7014

This theorem is referenced by:  pw1ninf  16935