Theorem 3nelsucpw1 7430

Description: Three is not an element of the successor of the power set of 1_o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
3nelsucpw1	⊢ ¬ 3o ∈ suc 𝒫 1o

Proof of Theorem 3nelsucpw1

Step	Hyp	Ref	Expression
1		1lt2o 6596	. . . . 5 ⊢ 1o ∈ 2o
2		elelsuc 4500	. . . . 5 ⊢ (1o ∈ 2o → 1o ∈ suc 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ∈ suc 2o
4		df-3o 6570	. . . 4 ⊢ 3o = suc 2o
5	3, 4	eleqtrri 2305	. . 3 ⊢ 1o ∈ 3o
6		ssnel 4661	. . 3 ⊢ (3o ⊆ 1o → ¬ 1o ∈ 3o)
7	5, 6	mt2 643	. 2 ⊢ ¬ 3o ⊆ 1o
8		pw1ne3 7426	. . . . . 6 ⊢ 𝒫 1o ≠ 3o
9	8	nesymi 2446	. . . . 5 ⊢ ¬ 3o = 𝒫 1o
10	9	a1i 9	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → ¬ 3o = 𝒫 1o)
11		elsuci 4494	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → (3o ∈ 𝒫 1o ∨ 3o = 𝒫 1o))
12	10, 11	ecased 1383	. . 3 ⊢ (3o ∈ suc 𝒫 1o → 3o ∈ 𝒫 1o)
13	12	elpwid 3660	. 2 ⊢ (3o ∈ suc 𝒫 1o → 3o ⊆ 1o)
14	7, 13	mto 666	1 ⊢ ¬ 3o ∈ suc 𝒫 1o

Syntax hints:  ¬ wn 3   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  𝒫 cpw 3649  suc csuc 4456  1_oc1o 6561  2_oc2o 6562  3_oc3o 6563

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 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  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 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-1o 6568  df-2o 6569  df-3o 6570

This theorem is referenced by:  onntri35  7433