Theorem sucpw1nel3 7414

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 6568	. . . . . . 7 ⊢ 1o ∈ V
2	1	pwex 4266	. . . . . 6 ⊢ 𝒫 1o ∈ V
3	2	sucid 4507	. . . . 5 ⊢ 𝒫 1o ∈ suc 𝒫 1o
4	3	ne0ii 3501	. . . 4 ⊢ suc 𝒫 1o ≠ ∅
5		pw1ne0 7409	. . . . . . . 8 ⊢ 𝒫 1o ≠ ∅
6	2	elsn 3682	. . . . . . . 8 ⊢ (𝒫 1o ∈ {∅} ↔ 𝒫 1o = ∅)
7	5, 6	nemtbir 2489	. . . . . . 7 ⊢ ¬ 𝒫 1o ∈ {∅}
8		df1o2 6573	. . . . . . . 8 ⊢ 1o = {∅}
9	8	eleq2i 2296	. . . . . . 7 ⊢ (𝒫 1o ∈ 1o ↔ 𝒫 1o ∈ {∅})
10	7, 9	mtbir 675	. . . . . 6 ⊢ ¬ 𝒫 1o ∈ 1o
11		eleq2 2293	. . . . . . 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 666	. . . . 5 ⊢ ¬ suc 𝒫 1o = 1o
14	13	neir 2403	. . . 4 ⊢ suc 𝒫 1o ≠ 1o
15	4, 14	nelpri 3690	. . 3 ⊢ ¬ suc 𝒫 1o ∈ {∅, 1o}
16		df2o3 6574	. . . 4 ⊢ 2o = {∅, 1o}
17	16	eleq2i 2296	. . 3 ⊢ (suc 𝒫 1o ∈ 2o ↔ suc 𝒫 1o ∈ {∅, 1o})
18	15, 17	mtbir 675	. 2 ⊢ ¬ suc 𝒫 1o ∈ 2o
19		pw1ne1 7410	. . . . . 6 ⊢ 𝒫 1o ≠ 1o
20	5, 19	nelpri 3690	. . . . 5 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
21	16	eleq2i 2296	. . . . 5 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
22	20, 21	mtbir 675	. . . 4 ⊢ ¬ 𝒫 1o ∈ 2o
23		eleq2 2293	. . . . 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 666	. . 3 ⊢ ¬ suc 𝒫 1o = 2o
26	2	sucex 4590	. . . 4 ⊢ suc 𝒫 1o ∈ V
27	26	elsn 3682	. . 3 ⊢ (suc 𝒫 1o ∈ {2o} ↔ suc 𝒫 1o = 2o)
28	25, 27	mtbir 675	. 2 ⊢ ¬ suc 𝒫 1o ∈ {2o}
29		ioran 757	. . 3 ⊢ (¬ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}) ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
30		df-3o 6562	. . . . . 6 ⊢ 3o = suc 2o
31		df-suc 4461	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
32	30, 31	eqtri 2250	. . . . 5 ⊢ 3o = (2o ∪ {2o})
33	32	eleq2i 2296	. . . 4 ⊢ (suc 𝒫 1o ∈ 3o ↔ suc 𝒫 1o ∈ (2o ∪ {2o}))
34		elun 3345	. . . 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 685	. 2 ⊢ (¬ suc 𝒫 1o ∈ 3o ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
37	18, 28, 36	mpbir2an 948	1 ⊢ ¬ suc 𝒫 1o ∈ 3o

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ∪ cun 3195  ∅c0 3491  𝒫 cpw 3649  {csn 3666  {cpr 3667  suc csuc 4455  1_oc1o 6553  2_oc2o 6554  3_oc3o 6555

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

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 3888  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-1o 6560  df-2o 6561  df-3o 6562

This theorem is referenced by:  onntri35  7418