Theorem pw1ninf 16814

Description: The powerset of 1_o is not infinite. Since we cannot prove it is finite (see pw1fin 7172), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7168. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ninf	⊢ ¬ ω ≼ 𝒫 1o

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16813	. 2 ⊢ ¬ 3o ≼ 𝒫 1o
2		3onn 6757	. . . 4 ⊢ 3o ∈ ω
3		nnfi 7129	. . . 4 ⊢ (3o ∈ ω → 3o ∈ Fin)
4		fict 7125	. . . 4 ⊢ (3o ∈ Fin → 3o ≼ ω)
5	2, 3, 4	mp2b 8	. . 3 ⊢ 3o ≼ ω
6		domtr 7027	. . 3 ⊢ ((3o ≼ ω ∧ ω ≼ 𝒫 1o) → 3o ≼ 𝒫 1o)
7	5, 6	mpan 424	. 2 ⊢ (ω ≼ 𝒫 1o → 3o ≼ 𝒫 1o)
8	1, 7	mto 668	1 ⊢ ¬ ω ≼ 𝒫 1o

Syntax hints:  ¬ wn 3   ∈ wcel 2205  𝒫 cpw 3671   class class class wbr 4111  ωcom 4714  1_oc1o 6642  3_oc3o 6644   ≼ cdom 6976  Fincfn 6977

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1o 6649  df-2o 6650  df-3o 6651  df-en 6978  df-dom 6979  df-fin 6980

This theorem is referenced by: (None)