Theorem 3nsssucpw1 7444

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 6579	. . . . . 6 ⊢ 3o = suc 2o
2	1	sseq1i 3251	. . . . 5 ⊢ (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o)
3		1lt2o 6605	. . . . . . . . 9 ⊢ 1o ∈ 2o
4		ssnel 4665	. . . . . . . . 9 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o)
5	3, 4	mt2 643	. . . . . . . 8 ⊢ ¬ 2o ⊆ 1o
6		2onn 6684	. . . . . . . . . 10 ⊢ 2o ∈ ω
7	6	elexi 2813	. . . . . . . . 9 ⊢ 2o ∈ V
8	7	elpw 3656	. . . . . . . 8 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o)
9	5, 8	mtbir 675	. . . . . . 7 ⊢ ¬ 2o ∈ 𝒫 1o
10	9	a1i 9	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → ¬ 2o ∈ 𝒫 1o)
11		sucssel 4519	. . . . . . . . 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 4498	. . . . . . . 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 734	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o = 𝒫 1o ∨ 2o ∈ 𝒫 1o))
16	10, 15	ecased 1383	. . . . 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 2235	. . 3 ⊢ (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o)
19		exmidpweq 7094	. . 3 ⊢ (EXMID ↔ 𝒫 1o = 2o)
20	18, 19	sylibr 134	. 2 ⊢ (3o ⊆ suc 𝒫 1o → EXMID)
21	20	con3i 635	1 ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ⊆ wss 3198  𝒫 cpw 3650  EXMIDwem 4282  suc csuc 4460  ωcom 4686  1_oc1o 6570  2_oc2o 6571  3_oc3o 6572

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-int 3927  df-tr 4186  df-exmid 4283  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-1o 6577  df-2o 6578  df-3o 6579

This theorem is referenced by:  onntri45  7449