Theorem 3nsssucpw1 7546

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 6649	. . . . . 6 ⊢ 3o = suc 2o
2	1	sseq1i 3264	. . . . 5 ⊢ (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o)
3		1lt2o 6675	. . . . . . . . 9 ⊢ 1o ∈ 2o
4		ssnel 4691	. . . . . . . . 9 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o)
5	3, 4	mt2 645	. . . . . . . 8 ⊢ ¬ 2o ⊆ 1o
6		2onn 6754	. . . . . . . . . 10 ⊢ 2o ∈ ω
7	6	elexi 2826	. . . . . . . . 9 ⊢ 2o ∈ V
8	7	elpw 3675	. . . . . . . 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 4545	. . . . . . . . 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 4524	. . . . . . . 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 2238	. . 3 ⊢ (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o)
19		exmidpweq 7169	. . 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 2203   ⊆ wss 3211  𝒫 cpw 3669  EXMIDwem 4307  suc csuc 4486  ωcom 4712  1_oc1o 6640  2_oc2o 6641  3_oc3o 6642

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

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 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-exmid 4308  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-1o 6647  df-2o 6648  df-3o 6649

This theorem is referenced by:  onntri45  7551