Theorem sucpw1nel3 7494

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 6633	. . . . . . 7 ⊢ 1o ∈ V
2	1	pwex 4279	. . . . . 6 ⊢ 𝒫 1o ∈ V
3	2	sucid 4520	. . . . 5 ⊢ 𝒫 1o ∈ suc 𝒫 1o
4	3	ne0ii 3506	. . . 4 ⊢ suc 𝒫 1o ≠ ∅
5		pw1ne0 7489	. . . . . . . 8 ⊢ 𝒫 1o ≠ ∅
6	2	elsn 3689	. . . . . . . 8 ⊢ (𝒫 1o ∈ {∅} ↔ 𝒫 1o = ∅)
7	5, 6	nemtbir 2492	. . . . . . 7 ⊢ ¬ 𝒫 1o ∈ {∅}
8		df1o2 6639	. . . . . . . 8 ⊢ 1o = {∅}
9	8	eleq2i 2298	. . . . . . 7 ⊢ (𝒫 1o ∈ 1o ↔ 𝒫 1o ∈ {∅})
10	7, 9	mtbir 678	. . . . . 6 ⊢ ¬ 𝒫 1o ∈ 1o
11		eleq2 2295	. . . . . . 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 668	. . . . 5 ⊢ ¬ suc 𝒫 1o = 1o
14	13	neir 2406	. . . 4 ⊢ suc 𝒫 1o ≠ 1o
15	4, 14	nelpri 3697	. . 3 ⊢ ¬ suc 𝒫 1o ∈ {∅, 1o}
16		df2o3 6640	. . . 4 ⊢ 2o = {∅, 1o}
17	16	eleq2i 2298	. . 3 ⊢ (suc 𝒫 1o ∈ 2o ↔ suc 𝒫 1o ∈ {∅, 1o})
18	15, 17	mtbir 678	. 2 ⊢ ¬ suc 𝒫 1o ∈ 2o
19		pw1ne1 7490	. . . . . 6 ⊢ 𝒫 1o ≠ 1o
20	5, 19	nelpri 3697	. . . . 5 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
21	16	eleq2i 2298	. . . . 5 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
22	20, 21	mtbir 678	. . . 4 ⊢ ¬ 𝒫 1o ∈ 2o
23		eleq2 2295	. . . . 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 668	. . 3 ⊢ ¬ suc 𝒫 1o = 2o
26	2	sucex 4603	. . . 4 ⊢ suc 𝒫 1o ∈ V
27	26	elsn 3689	. . 3 ⊢ (suc 𝒫 1o ∈ {2o} ↔ suc 𝒫 1o = 2o)
28	25, 27	mtbir 678	. 2 ⊢ ¬ suc 𝒫 1o ∈ {2o}
29		ioran 760	. . 3 ⊢ (¬ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}) ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
30		df-3o 6627	. . . . . 6 ⊢ 3o = suc 2o
31		df-suc 4474	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
32	30, 31	eqtri 2252	. . . . 5 ⊢ 3o = (2o ∪ {2o})
33	32	eleq2i 2298	. . . 4 ⊢ (suc 𝒫 1o ∈ 3o ↔ suc 𝒫 1o ∈ (2o ∪ {2o}))
34		elun 3350	. . . 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 688	. 2 ⊢ (¬ suc 𝒫 1o ∈ 3o ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
37	18, 28, 36	mpbir2an 951	1 ⊢ ¬ suc 𝒫 1o ∈ 3o

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2202   ∪ cun 3199  ∅c0 3496  𝒫 cpw 3656  {csn 3673  {cpr 3674  suc csuc 4468  1_oc1o 6618  2_oc2o 6619  3_oc3o 6620

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-1o 6625  df-2o 6626  df-3o 6627

This theorem is referenced by:  onntri35  7498