Theorem pw1nel3 7291

Description: Negated excluded middle implies that the power set of 1_o is not an element of 3_o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
pw1nel3	⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)

Proof of Theorem pw1nel3

Step	Hyp	Ref	Expression
1		pw1ne0 7288	. . . . 5 ⊢ 𝒫 1o ≠ ∅
2		pw1ne1 7289	. . . . 5 ⊢ 𝒫 1o ≠ 1o
3	1, 2	nelpri 3642	. . . 4 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
4	3	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5		df2o3 6483	. . . 4 ⊢ 2o = {∅, 1o}
6	5	eleq2i 2260	. . 3 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
7	4, 6	sylnibr 678	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 2o)
8		exmidpweq 6965	. . . 4 ⊢ (EXMID ↔ 𝒫 1o = 2o)
9	8	notbii 669	. . 3 ⊢ (¬ EXMID ↔ ¬ 𝒫 1o = 2o)
10		1oex 6477	. . . . . 6 ⊢ 1o ∈ V
11	10	pwex 4212	. . . . 5 ⊢ 𝒫 1o ∈ V
12	11	elsn 3634	. . . 4 ⊢ (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
13	12	notbii 669	. . 3 ⊢ (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
14	9, 13	sylbb2 138	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {2o})
15		df-3o 6471	. . . . . . 7 ⊢ 3o = suc 2o
16		df-suc 4402	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
17	15, 16	eqtri 2214	. . . . . 6 ⊢ 3o = (2o ∪ {2o})
18	17	eleq2i 2260	. . . . 5 ⊢ (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19		elun 3300	. . . . 5 ⊢ (𝒫 1o ∈ (2o ∪ {2o}) ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
20	18, 19	bitri 184	. . . 4 ⊢ (𝒫 1o ∈ 3o ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
21	20	notbii 669	. . 3 ⊢ (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22		ioran 753	. . 3 ⊢ (¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}) ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
23	21, 22	bitri 184	. 2 ⊢ (¬ 𝒫 1o ∈ 3o ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
24	7, 14, 23	sylanbrc 417	1 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164   ∪ cun 3151  ∅c0 3446  𝒫 cpw 3601  {csn 3618  {cpr 3619  EXMIDwem 4223  suc csuc 4396  1_oc1o 6462  2_oc2o 6463  3_oc3o 6464

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-exmid 4224  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469  df-2o 6470  df-3o 6471

This theorem is referenced by:  sucpw1ne3  7292  sucpw1nss3  7295