Theorem pw1ninf 16611

Description: The powerset of 1_o is not infinite. Since we cannot prove it is finite (see pw1fin 7102), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7098. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ninf	⊢ ¬ ω ≼ 𝒫 1o

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16610	. 2 ⊢ ¬ 3o ≼ 𝒫 1o
2		3onn 6690	. . . 4 ⊢ 3o ∈ ω
3		nnfi 7059	. . . 4 ⊢ (3o ∈ ω → 3o ∈ Fin)
4		fict 7055	. . . 4 ⊢ (3o ∈ Fin → 3o ≼ ω)
5	2, 3, 4	mp2b 8	. . 3 ⊢ 3o ≼ ω
6		domtr 6959	. . 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 2202  𝒫 cpw 3652   class class class wbr 4088  ωcom 4688  1_oc1o 6575  3_oc3o 6577   ≼ cdom 6908  Fincfn 6909

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6582  df-2o 6583  df-3o 6584  df-en 6910  df-dom 6911  df-fin 6912

This theorem is referenced by: (None)