Theorem pw1nel3 7232

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 7229	. . . . 5 ⊢ 𝒫 1o ≠ ∅
2		pw1ne1 7230	. . . . 5 ⊢ 𝒫 1o ≠ 1o
3	1, 2	nelpri 3618	. . . 4 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
4	3	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5		df2o3 6433	. . . 4 ⊢ 2o = {∅, 1o}
6	5	eleq2i 2244	. . 3 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
7	4, 6	sylnibr 677	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 2o)
8		exmidpweq 6911	. . . 4 ⊢ (EXMID ↔ 𝒫 1o = 2o)
9	8	notbii 668	. . 3 ⊢ (¬ EXMID ↔ ¬ 𝒫 1o = 2o)
10		1oex 6427	. . . . . 6 ⊢ 1o ∈ V
11	10	pwex 4185	. . . . 5 ⊢ 𝒫 1o ∈ V
12	11	elsn 3610	. . . 4 ⊢ (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
13	12	notbii 668	. . 3 ⊢ (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
14	9, 13	sylbb2 138	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {2o})
15		df-3o 6421	. . . . . . 7 ⊢ 3o = suc 2o
16		df-suc 4373	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
17	15, 16	eqtri 2198	. . . . . 6 ⊢ 3o = (2o ∪ {2o})
18	17	eleq2i 2244	. . . . 5 ⊢ (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19		elun 3278	. . . . 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 668	. . 3 ⊢ (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22		ioran 752	. . 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 708   = wceq 1353   ∈ wcel 2148   ∪ cun 3129  ∅c0 3424  𝒫 cpw 3577  {csn 3594  {cpr 3595  EXMIDwem 4196  suc csuc 4367  1_oc1o 6412  2_oc2o 6413  3_oc3o 6414

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373  df-1o 6419  df-2o 6420  df-3o 6421

This theorem is referenced by:  sucpw1ne3  7233  sucpw1nss3  7236