Theorem pw1nel3 7208

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 7205	. . . . 5 ⊢ 𝒫 1o ≠ ∅
2		pw1ne1 7206	. . . . 5 ⊢ 𝒫 1o ≠ 1o
3	1, 2	nelpri 3607	. . . 4 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
4	3	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5		df2o3 6409	. . . 4 ⊢ 2o = {∅, 1o}
6	5	eleq2i 2237	. . 3 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
7	4, 6	sylnibr 672	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 2o)
8		exmidpweq 6887	. . . 4 ⊢ (EXMID ↔ 𝒫 1o = 2o)
9	8	notbii 663	. . 3 ⊢ (¬ EXMID ↔ ¬ 𝒫 1o = 2o)
10		1oex 6403	. . . . . 6 ⊢ 1o ∈ V
11	10	pwex 4169	. . . . 5 ⊢ 𝒫 1o ∈ V
12	11	elsn 3599	. . . 4 ⊢ (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
13	12	notbii 663	. . 3 ⊢ (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
14	9, 13	sylbb2 137	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {2o})
15		df-3o 6397	. . . . . . 7 ⊢ 3o = suc 2o
16		df-suc 4356	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
17	15, 16	eqtri 2191	. . . . . 6 ⊢ 3o = (2o ∪ {2o})
18	17	eleq2i 2237	. . . . 5 ⊢ (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19		elun 3268	. . . . 5 ⊢ (𝒫 1o ∈ (2o ∪ {2o}) ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
20	18, 19	bitri 183	. . . 4 ⊢ (𝒫 1o ∈ 3o ↔ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
21	20	notbii 663	. . 3 ⊢ (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22		ioran 747	. . 3 ⊢ (¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}) ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
23	21, 22	bitri 183	. 2 ⊢ (¬ 𝒫 1o ∈ 3o ↔ (¬ 𝒫 1o ∈ 2o ∧ ¬ 𝒫 1o ∈ {2o}))
24	7, 14, 23	sylanbrc 415	1 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141   ∪ cun 3119  ∅c0 3414  𝒫 cpw 3566  {csn 3583  {cpr 3584  EXMIDwem 4180  suc csuc 4350  1_oc1o 6388  2_oc2o 6389  3_oc3o 6390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356  df-1o 6395  df-2o 6396  df-3o 6397

This theorem is referenced by:  sucpw1ne3  7209  sucpw1nss3  7212