Theorem pw1ninf 16765

Description: The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7170), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7166. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

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

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16764	. 2 |- -. 3o ~<_ ~P 1o
2		3onn 6755	. . . 4 |- 3o e. om
3		nnfi 7127	. . . 4 |- ( 3o e. om -> 3o e. Fin )
4		fict 7123	. . . 4 |- ( 3o e. Fin -> 3o ~<_ om )
5	2, 3, 4	mp2b 8	. . 3 |- 3o ~<_ om
6		domtr 7025	. . 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 2203   ~Pcpw 3669   class class class wbr 4109   omcom 4712   1oc1o 6640   3oc3o 6642   ~<_ cdom 6974   Fincfn 6975

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-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-2o 6648  df-3o 6649  df-en 6976  df-dom 6977  df-fin 6978

This theorem is referenced by: (None)