Theorem 3nsssucpw1 7559

Description: Negated excluded middle implies that 3_o is not a subset of the successor of the power set of 1_o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)

Assertion

Ref	Expression
3nsssucpw1	⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Proof of Theorem 3nsssucpw1

Step	Hyp	Ref	Expression
1		df-3o 6662	. . . . . 6 ⊢ 3o = suc 2o
2	1	sseq1i 3268	. . . . 5 ⊢ (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o)
3		1lt2o 6688	. . . . . . . . 9 ⊢ 1o ∈ 2o
4		ssnel 4696	. . . . . . . . 9 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o)
5	3, 4	mt2 645	. . . . . . . 8 ⊢ ¬ 2o ⊆ 1o
6		2onn 6767	. . . . . . . . . 10 ⊢ 2o ∈ ω
7	6	elexi 2828	. . . . . . . . 9 ⊢ 2o ∈ V
8	7	elpw 3680	. . . . . . . 8 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o)
9	5, 8	mtbir 678	. . . . . . 7 ⊢ ¬ 2o ∈ 𝒫 1o
10	9	a1i 9	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → ¬ 2o ∈ 𝒫 1o)
11		sucssel 4550	. . . . . . . . 9 ⊢ (2o ∈ ω → (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o))
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ (suc 2o ⊆ suc 𝒫 1o → 2o ∈ suc 𝒫 1o)
13		elsuci 4529	. . . . . . . 8 ⊢ (2o ∈ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o))
14	12, 13	syl 14	. . . . . . 7 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o ∈ 𝒫 1o ∨ 2o = 𝒫 1o))
15	14	orcomd 737	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o = 𝒫 1o ∨ 2o ∈ 𝒫 1o))
16	10, 15	ecased 1386	. . . . 5 ⊢ (suc 2o ⊆ suc 𝒫 1o → 2o = 𝒫 1o)
17	2, 16	sylbi 121	. . . 4 ⊢ (3o ⊆ suc 𝒫 1o → 2o = 𝒫 1o)
18	17	eqcomd 2240	. . 3 ⊢ (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o)
19		exmidpweq 7182	. . 3 ⊢ (EXMID ↔ 𝒫 1o = 2o)
20	18, 19	sylibr 134	. 2 ⊢ (3o ⊆ suc 𝒫 1o → EXMID)
21	20	con3i 637	1 ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 716   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  𝒫 cpw 3674  EXMIDwem 4312  suc csuc 4491  ωcom 4717  1_oc1o 6653  2_oc2o 6654  3_oc3o 6655

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-tr 4214  df-exmid 4313  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  onntri45  7564