Theorem pw1nel3 7362

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 7359	. . . . 5 ⊢ 𝒫 1o ≠ ∅
2		pw1ne1 7360	. . . . 5 ⊢ 𝒫 1o ≠ 1o
3	1, 2	nelpri 3662	. . . 4 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
4	3	a1i 9	. . 3 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {∅, 1o})
5		df2o3 6529	. . . 4 ⊢ 2o = {∅, 1o}
6	5	eleq2i 2273	. . 3 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
7	4, 6	sylnibr 679	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 2o)
8		exmidpweq 7021	. . . 4 ⊢ (EXMID ↔ 𝒫 1o = 2o)
9	8	notbii 670	. . 3 ⊢ (¬ EXMID ↔ ¬ 𝒫 1o = 2o)
10		1oex 6523	. . . . . 6 ⊢ 1o ∈ V
11	10	pwex 4235	. . . . 5 ⊢ 𝒫 1o ∈ V
12	11	elsn 3654	. . . 4 ⊢ (𝒫 1o ∈ {2o} ↔ 𝒫 1o = 2o)
13	12	notbii 670	. . 3 ⊢ (¬ 𝒫 1o ∈ {2o} ↔ ¬ 𝒫 1o = 2o)
14	9, 13	sylbb2 138	. 2 ⊢ (¬ EXMID → ¬ 𝒫 1o ∈ {2o})
15		df-3o 6517	. . . . . . 7 ⊢ 3o = suc 2o
16		df-suc 4426	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
17	15, 16	eqtri 2227	. . . . . 6 ⊢ 3o = (2o ∪ {2o})
18	17	eleq2i 2273	. . . . 5 ⊢ (𝒫 1o ∈ 3o ↔ 𝒫 1o ∈ (2o ∪ {2o}))
19		elun 3318	. . . . 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 670	. . 3 ⊢ (¬ 𝒫 1o ∈ 3o ↔ ¬ (𝒫 1o ∈ 2o ∨ 𝒫 1o ∈ {2o}))
22		ioran 754	. . 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 710   = wceq 1373   ∈ wcel 2177   ∪ cun 3168  ∅c0 3464  𝒫 cpw 3621  {csn 3638  {cpr 3639  EXMIDwem 4246  suc csuc 4420  1_oc1o 6508  2_oc2o 6509  3_oc3o 6510

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-tr 4151  df-exmid 4247  df-iord 4421  df-on 4423  df-suc 4426  df-1o 6515  df-2o 6516  df-3o 6517

This theorem is referenced by:  sucpw1ne3  7363  sucpw1nss3  7366