Theorem sucpw1nel3 7556

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 6668	. . . . . . 7 ⊢ 1o ∈ V
2	1	pwex 4301	. . . . . 6 ⊢ 𝒫 1o ∈ V
3	2	sucid 4543	. . . . 5 ⊢ 𝒫 1o ∈ suc 𝒫 1o
4	3	ne0ii 3522	. . . 4 ⊢ suc 𝒫 1o ≠ ∅
5		pw1ne0 7551	. . . . . . . 8 ⊢ 𝒫 1o ≠ ∅
6	2	elsn 3710	. . . . . . . 8 ⊢ (𝒫 1o ∈ {∅} ↔ 𝒫 1o = ∅)
7	5, 6	nemtbir 2503	. . . . . . 7 ⊢ ¬ 𝒫 1o ∈ {∅}
8		df1o2 6674	. . . . . . . 8 ⊢ 1o = {∅}
9	8	eleq2i 2301	. . . . . . 7 ⊢ (𝒫 1o ∈ 1o ↔ 𝒫 1o ∈ {∅})
10	7, 9	mtbir 678	. . . . . 6 ⊢ ¬ 𝒫 1o ∈ 1o
11		eleq2 2298	. . . . . . 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 2417	. . . 4 ⊢ suc 𝒫 1o ≠ 1o
15	4, 14	nelpri 3718	. . 3 ⊢ ¬ suc 𝒫 1o ∈ {∅, 1o}
16		df2o3 6675	. . . 4 ⊢ 2o = {∅, 1o}
17	16	eleq2i 2301	. . 3 ⊢ (suc 𝒫 1o ∈ 2o ↔ suc 𝒫 1o ∈ {∅, 1o})
18	15, 17	mtbir 678	. 2 ⊢ ¬ suc 𝒫 1o ∈ 2o
19		pw1ne1 7552	. . . . . 6 ⊢ 𝒫 1o ≠ 1o
20	5, 19	nelpri 3718	. . . . 5 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
21	16	eleq2i 2301	. . . . 5 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
22	20, 21	mtbir 678	. . . 4 ⊢ ¬ 𝒫 1o ∈ 2o
23		eleq2 2298	. . . . 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 4626	. . . 4 ⊢ suc 𝒫 1o ∈ V
27	26	elsn 3710	. . 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 6662	. . . . . 6 ⊢ 3o = suc 2o
31		df-suc 4497	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
32	30, 31	eqtri 2255	. . . . 5 ⊢ 3o = (2o ∪ {2o})
33	32	eleq2i 2301	. . . 4 ⊢ (suc 𝒫 1o ∈ 3o ↔ suc 𝒫 1o ∈ (2o ∪ {2o}))
34		elun 3364	. . . 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 2205   ∪ cun 3212  ∅c0 3512  𝒫 cpw 3674  {csn 3694  {cpr 3695  suc csuc 4491  1_oc1o 6653  2_oc2o 6654  3_oc3o 6655

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  onntri35  7560