Theorem pw1ninf 16384

Description: The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7080), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7076. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ninf	|- -. om ~<_ ~P 1o

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16383	. 2 |- -. 3o ~<_ ~P 1o
2		3onn 6676	. . . 4 |- 3o e. om
3		nnfi 7042	. . . 4 |- ( 3o e. om -> 3o e. Fin )
4		fict 7038	. . . 4 |- ( 3o e. Fin -> 3o ~<_ om )
5	2, 3, 4	mp2b 8	. . 3 |- 3o ~<_ om
6		domtr 6945	. . 3 |- ( ( 3o ~<_ om /\ om ~<_ ~P 1o ) -> 3o ~<_ ~P 1o )
7	5, 6	mpan 424	. 2 |- ( om ~<_ ~P 1o -> 3o ~<_ ~P 1o )
8	1, 7	mto 666	1 |- -. om ~<_ ~P 1o

Syntax hints:   -. wn 3   e. wcel 2200   ~Pcpw 3649   class class class wbr 4083   omcom 4682   1oc1o 6561   3oc3o 6563   ~<_ cdom 6894   Fincfn 6895

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-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6568  df-2o 6569  df-3o 6570  df-en 6896  df-dom 6897  df-fin 6898

This theorem is referenced by: (None)