Theorem pw1ninf 16688

Description: The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7145), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7141. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

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

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16687	. 2 |- -. 3o ~<_ ~P 1o
2		3onn 6733	. . . 4 |- 3o e. om
3		nnfi 7102	. . . 4 |- ( 3o e. om -> 3o e. Fin )
4		fict 7098	. . . 4 |- ( 3o e. Fin -> 3o ~<_ om )
5	2, 3, 4	mp2b 8	. . 3 |- 3o ~<_ om
6		domtr 7002	. . 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 668	1 |- -. om ~<_ ~P 1o

Syntax hints:   -. wn 3   e. wcel 2202   ~Pcpw 3656   class class class wbr 4093   omcom 4694   1oc1o 6618   3oc3o 6620   ~<_ cdom 6951   Fincfn 6952

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-3o 6627  df-en 6953  df-dom 6954  df-fin 6955

This theorem is referenced by: (None)