Theorem sucpw1nss3 7408

Description: Negated excluded middle implies that the successor of the power set of 1_o is not a subset of 3_o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)

Assertion

Ref	Expression
sucpw1nss3	⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)

Proof of Theorem sucpw1nss3

Step	Hyp	Ref	Expression
1		pw1nel3 7404	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)
2		pw1on 7399	. . 3 ⊢ 𝒫 1o ∈ On
3		sucssel 4512	. . 3 ⊢ (𝒫 1o ∈ On → (suc 𝒫 1o ⊆ 3o → 𝒫 1o ∈ 3o))
4	2, 3	ax-mp 5	. 2 ⊢ (suc 𝒫 1o ⊆ 3o → 𝒫 1o ∈ 3o)
5	1, 4	nsyl 631	1 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2200   ⊆ wss 3197  𝒫 cpw 3649  EXMIDwem 4277  Oncon0 4451  suc csuc 4453  1_oc1o 6545  3_oc3o 6547

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-exmid 4278  df-iord 4454  df-on 4456  df-suc 4459  df-1o 6552  df-2o 6553  df-3o 6554

This theorem is referenced by:  onntri45  7414