Theorem pw1ninf 17004

Description: The powerset of 1_o is not infinite. Since we cannot prove it is finite (see pw1fin 7211), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7207. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ninf	⊢ ¬ ω ≼ 𝒫 1o

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 17003	. 2 ⊢ ¬ 3o ≼ 𝒫 1o
2		3onn 6789	. . . 4 ⊢ 3o ∈ ω
3		nnfi 7168	. . . 4 ⊢ (3o ∈ ω → 3o ∈ Fin)
4		fict 7164	. . . 4 ⊢ (3o ∈ Fin → 3o ≼ ω)
5	2, 3, 4	mp2b 8	. . 3 ⊢ 3o ≼ ω
6		domtr 7066	. . 3 ⊢ ((3o ≼ ω ∧ ω ≼ 𝒫 1o) → 3o ≼ 𝒫 1o)
7	5, 6	mpan 428	. 2 ⊢ (ω ≼ 𝒫 1o → 3o ≼ 𝒫 1o)
8	1, 7	mto 672	1 ⊢ ¬ ω ≼ 𝒫 1o

Syntax hints:  ¬ wn 3   ∈ wcel 2209  𝒫 cpw 3688   class class class wbr 4128  ωcom 4735  1_oc1o 6674  3_oc3o 6676   ≼ cdom 7015  Fincfn 7016

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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-2o 6682  df-3o 6683  df-en 7017  df-dom 7018  df-fin 7019

This theorem is referenced by: (None)