Theorem pw1ninf 16650

Description: The powerset of 1_o is not infinite. Since we cannot prove it is finite (see pw1fin 7107), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7103. (Contributed by Jim Kingdon, 14-Feb-2026.)

Assertion

Ref	Expression
pw1ninf	⊢ ¬ ω ≼ 𝒫 1o

Proof of Theorem pw1ninf

Step	Hyp	Ref	Expression
1		pw1ndom3 16649	. 2 ⊢ ¬ 3o ≼ 𝒫 1o
2		3onn 6695	. . . 4 ⊢ 3o ∈ ω
3		nnfi 7064	. . . 4 ⊢ (3o ∈ ω → 3o ∈ Fin)
4		fict 7060	. . . 4 ⊢ (3o ∈ Fin → 3o ≼ ω)
5	2, 3, 4	mp2b 8	. . 3 ⊢ 3o ≼ ω
6		domtr 6964	. . 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 2201  𝒫 cpw 3653   class class class wbr 4089  ωcom 4690  1_oc1o 6580  3_oc3o 6582   ≼ cdom 6913  Fincfn 6914

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1o 6587  df-2o 6588  df-3o 6589  df-en 6915  df-dom 6916  df-fin 6917

This theorem is referenced by: (None)