Theorem sucpw1nel3 7295

Description: The successor of the power set of 1_o is not an element of 3_o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
sucpw1nel3	⊢ ¬ suc 𝒫 1o ∈ 3o

Proof of Theorem sucpw1nel3

Step	Hyp	Ref	Expression
1		1oex 6479	. . . . . . 7 ⊢ 1o ∈ V
2	1	pwex 4213	. . . . . 6 ⊢ 𝒫 1o ∈ V
3	2	sucid 4449	. . . . 5 ⊢ 𝒫 1o ∈ suc 𝒫 1o
4	3	ne0ii 3457	. . . 4 ⊢ suc 𝒫 1o ≠ ∅
5		pw1ne0 7290	. . . . . . . 8 ⊢ 𝒫 1o ≠ ∅
6	2	elsn 3635	. . . . . . . 8 ⊢ (𝒫 1o ∈ {∅} ↔ 𝒫 1o = ∅)
7	5, 6	nemtbir 2453	. . . . . . 7 ⊢ ¬ 𝒫 1o ∈ {∅}
8		df1o2 6484	. . . . . . . 8 ⊢ 1o = {∅}
9	8	eleq2i 2260	. . . . . . 7 ⊢ (𝒫 1o ∈ 1o ↔ 𝒫 1o ∈ {∅})
10	7, 9	mtbir 672	. . . . . 6 ⊢ ¬ 𝒫 1o ∈ 1o
11		eleq2 2257	. . . . . . 7 ⊢ (suc 𝒫 1o = 1o → (𝒫 1o ∈ suc 𝒫 1o ↔ 𝒫 1o ∈ 1o))
12	3, 11	mpbii 148	. . . . . 6 ⊢ (suc 𝒫 1o = 1o → 𝒫 1o ∈ 1o)
13	10, 12	mto 663	. . . . 5 ⊢ ¬ suc 𝒫 1o = 1o
14	13	neir 2367	. . . 4 ⊢ suc 𝒫 1o ≠ 1o
15	4, 14	nelpri 3643	. . 3 ⊢ ¬ suc 𝒫 1o ∈ {∅, 1o}
16		df2o3 6485	. . . 4 ⊢ 2o = {∅, 1o}
17	16	eleq2i 2260	. . 3 ⊢ (suc 𝒫 1o ∈ 2o ↔ suc 𝒫 1o ∈ {∅, 1o})
18	15, 17	mtbir 672	. 2 ⊢ ¬ suc 𝒫 1o ∈ 2o
19		pw1ne1 7291	. . . . . 6 ⊢ 𝒫 1o ≠ 1o
20	5, 19	nelpri 3643	. . . . 5 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
21	16	eleq2i 2260	. . . . 5 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
22	20, 21	mtbir 672	. . . 4 ⊢ ¬ 𝒫 1o ∈ 2o
23		eleq2 2257	. . . . 5 ⊢ (suc 𝒫 1o = 2o → (𝒫 1o ∈ suc 𝒫 1o ↔ 𝒫 1o ∈ 2o))
24	3, 23	mpbii 148	. . . 4 ⊢ (suc 𝒫 1o = 2o → 𝒫 1o ∈ 2o)
25	22, 24	mto 663	. . 3 ⊢ ¬ suc 𝒫 1o = 2o
26	2	sucex 4532	. . . 4 ⊢ suc 𝒫 1o ∈ V
27	26	elsn 3635	. . 3 ⊢ (suc 𝒫 1o ∈ {2o} ↔ suc 𝒫 1o = 2o)
28	25, 27	mtbir 672	. 2 ⊢ ¬ suc 𝒫 1o ∈ {2o}
29		ioran 753	. . 3 ⊢ (¬ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}) ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
30		df-3o 6473	. . . . . 6 ⊢ 3o = suc 2o
31		df-suc 4403	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
32	30, 31	eqtri 2214	. . . . 5 ⊢ 3o = (2o ∪ {2o})
33	32	eleq2i 2260	. . . 4 ⊢ (suc 𝒫 1o ∈ 3o ↔ suc 𝒫 1o ∈ (2o ∪ {2o}))
34		elun 3301	. . . 4 ⊢ (suc 𝒫 1o ∈ (2o ∪ {2o}) ↔ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}))
35	33, 34	bitri 184	. . 3 ⊢ (suc 𝒫 1o ∈ 3o ↔ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}))
36	29, 35	xchnxbir 682	. 2 ⊢ (¬ suc 𝒫 1o ∈ 3o ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
37	18, 28, 36	mpbir2an 944	1 ⊢ ¬ suc 𝒫 1o ∈ 3o

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164   ∪ cun 3152  ∅c0 3447  𝒫 cpw 3602  {csn 3619  {cpr 3620  suc csuc 4397  1_oc1o 6464  2_oc2o 6465  3_oc3o 6466

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403  df-1o 6471  df-2o 6472  df-3o 6473

This theorem is referenced by:  onntri35  7299