Theorem 3nsssucpw1 7585

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 6679	. . . . . 6 ⊢ 3o = suc 2o
2	1	sseq1i 3274	. . . . 5 ⊢ (3o ⊆ suc 𝒫 1o ↔ suc 2o ⊆ suc 𝒫 1o)
3		1lt2o 6705	. . . . . . . . 9 ⊢ 1o ∈ 2o
4		ssnel 4711	. . . . . . . . 9 ⊢ (2o ⊆ 1o → ¬ 1o ∈ 2o)
5	3, 4	mt2 649	. . . . . . . 8 ⊢ ¬ 2o ⊆ 1o
6		2onn 6784	. . . . . . . . . 10 ⊢ 2o ∈ ω
7	6	elexi 2834	. . . . . . . . 9 ⊢ 2o ∈ V
8	7	elpw 3691	. . . . . . . 8 ⊢ (2o ∈ 𝒫 1o ↔ 2o ⊆ 1o)
9	5, 8	mtbir 682	. . . . . . 7 ⊢ ¬ 2o ∈ 𝒫 1o
10	9	a1i 9	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → ¬ 2o ∈ 𝒫 1o)
11		sucssel 4564	. . . . . . . . 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 4543	. . . . . . . 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 741	. . . . . 6 ⊢ (suc 2o ⊆ suc 𝒫 1o → (2o = 𝒫 1o ∨ 2o ∈ 𝒫 1o))
16	10, 15	ecased 1390	. . . . 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 2244	. . 3 ⊢ (3o ⊆ suc 𝒫 1o → 𝒫 1o = 2o)
19		exmidpweq 7206	. . 3 ⊢ (EXMID ↔ 𝒫 1o = 2o)
20	18, 19	sylibr 134	. 2 ⊢ (3o ⊆ suc 𝒫 1o → EXMID)
21	20	con3i 641	1 ⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  𝒫 cpw 3685  EXMIDwem 4326  suc csuc 4505  ωcom 4732  1_oc1o 6670  2_oc2o 6671  3_oc3o 6672

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-1o 6677  df-2o 6678  df-3o 6679

This theorem is referenced by:  onntri45  7590