Theorem pw1nel3 7160

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 7157	. . . . 5 ⊢ 𝒫 1o ≠ ∅
2		pw1ne1 7158	. . . . 5 ⊢ 𝒫 1o ≠ 1o
3	1, 2	nelpri 3584	. . . 4 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
4	3	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5		df2o3 6374	. . . 4 ⊢ 2o = {∅, 1o}
6	5	eleq2i 2224	. . 3 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
7	4, 6	sylnibr 667	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 2o)
8		exmidpweq 6851	. . . 4 ⊢ (EXMID ↔ 𝒫 1o = 2o)
9	8	notbii 658	. . 3 ⊢ (¬ EXMID ↔ ¬ 𝒫 1o = 2o)
10		1oex 6368	. . . . . 6 ⊢ 1o ∈ V
11	10	pwex 4144	. . . . 5 ⊢ 𝒫 1o ∈ V
12	11	elsn 3576	. . . 4 ⊢ (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
13	12	notbii 658	. . 3 ⊢ (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
14	9, 13	sylbb2 137	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {2o})
15		df-3o 6362	. . . . . . 7 ⊢ 3o = suc 2o
16		df-suc 4331	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
17	15, 16	eqtri 2178	. . . . . 6 ⊢ 3o = (2o ∪ {2o})
18	17	eleq2i 2224	. . . . 5 ⊢ (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19		elun 3248	. . . . 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 658	. . 3 ⊢ (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22		ioran 742	. . 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 414	1 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1335   ∈ wcel 2128   ∪ cun 3100  ∅c0 3394  𝒫 cpw 3543  {csn 3560  {cpr 3561  EXMIDwem 4155  suc csuc 4325  1_oc1o 6353  2_oc2o 6354  3_oc3o 6355

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331  df-1o 6360  df-2o 6361  df-3o 6362

This theorem is referenced by:  sucpw1ne3  7161  sucpw1nss3  7164