Theorem sucpw1nss3 7381

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 7377	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)
2		pw1on 7372	. . 3 ⊢ 𝒫 1o ∈ On
3		sucssel 4489	. . 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 629	1 ⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2178   ⊆ wss 3174  𝒫 cpw 3626  EXMIDwem 4254  Oncon0 4428  suc csuc 4430  1_oc1o 6518  3_oc3o 6520

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-exmid 4255  df-iord 4431  df-on 4433  df-suc 4436  df-1o 6525  df-2o 6526  df-3o 6527

This theorem is referenced by:  onntri45  7387