Theorem 3nelsucpw1 7544

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 6675	. . . . 5 ⊢ 1o ∈ 2o
2		elelsuc 4530	. . . . 5 ⊢ (1o ∈ 2o → 1o ∈ suc 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ∈ suc 2o
4		df-3o 6649	. . . 4 ⊢ 3o = suc 2o
5	3, 4	eleqtrri 2308	. . 3 ⊢ 1o ∈ 3o
6		ssnel 4691	. . 3 ⊢ (3o ⊆ 1o → ¬ 1o ∈ 3o)
7	5, 6	mt2 645	. 2 ⊢ ¬ 3o ⊆ 1o
8		pw1ne3 7540	. . . . . 6 ⊢ 𝒫 1o ≠ 3o
9	8	nesymi 2458	. . . . 5 ⊢ ¬ 3o = 𝒫 1o
10	9	a1i 9	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → ¬ 3o = 𝒫 1o)
11		elsuci 4524	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → (3o ∈ 𝒫 1o ∨ 3o = 𝒫 1o))
12	10, 11	ecased 1386	. . 3 ⊢ (3o ∈ suc 𝒫 1o → 3o ∈ 𝒫 1o)
13	12	elpwid 3680	. 2 ⊢ (3o ∈ suc 𝒫 1o → 3o ⊆ 1o)
14	7, 13	mto 668	1 ⊢ ¬ 3o ∈ suc 𝒫 1o

Syntax hints:  ¬ wn 3   = wceq 1398   ∈ wcel 2203   ⊆ wss 3211  𝒫 cpw 3669  suc csuc 4486  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-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-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:  onntri35  7547