Theorem sucpw1nel3 7582

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 6685	. . . . . . 7 ⊢ 1o ∈ V
2	1	pwex 4315	. . . . . 6 ⊢ 𝒫 1o ∈ V
3	2	sucid 4557	. . . . 5 ⊢ 𝒫 1o ∈ suc 𝒫 1o
4	3	ne0ii 3531	. . . 4 ⊢ suc 𝒫 1o ≠ ∅
5		pw1ne0 7577	. . . . . . . 8 ⊢ 𝒫 1o ≠ ∅
6	2	elsn 3721	. . . . . . . 8 ⊢ (𝒫 1o ∈ {∅} ↔ 𝒫 1o = ∅)
7	5, 6	nemtbir 2509	. . . . . . 7 ⊢ ¬ 𝒫 1o ∈ {∅}
8		df1o2 6691	. . . . . . . 8 ⊢ 1o = {∅}
9	8	eleq2i 2305	. . . . . . 7 ⊢ (𝒫 1o ∈ 1o ↔ 𝒫 1o ∈ {∅})
10	7, 9	mtbir 682	. . . . . 6 ⊢ ¬ 𝒫 1o ∈ 1o
11		eleq2 2302	. . . . . . 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 672	. . . . 5 ⊢ ¬ suc 𝒫 1o = 1o
14	13	neir 2423	. . . 4 ⊢ suc 𝒫 1o ≠ 1o
15	4, 14	nelpri 3729	. . 3 ⊢ ¬ suc 𝒫 1o ∈ {∅, 1o}
16		df2o3 6692	. . . 4 ⊢ 2o = {∅, 1o}
17	16	eleq2i 2305	. . 3 ⊢ (suc 𝒫 1o ∈ 2o ↔ suc 𝒫 1o ∈ {∅, 1o})
18	15, 17	mtbir 682	. 2 ⊢ ¬ suc 𝒫 1o ∈ 2o
19		pw1ne1 7578	. . . . . 6 ⊢ 𝒫 1o ≠ 1o
20	5, 19	nelpri 3729	. . . . 5 ⊢ ¬ 𝒫 1o ∈ {∅, 1o}
21	16	eleq2i 2305	. . . . 5 ⊢ (𝒫 1o ∈ 2o ↔ 𝒫 1o ∈ {∅, 1o})
22	20, 21	mtbir 682	. . . 4 ⊢ ¬ 𝒫 1o ∈ 2o
23		eleq2 2302	. . . . 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 672	. . 3 ⊢ ¬ suc 𝒫 1o = 2o
26	2	sucex 4641	. . . 4 ⊢ suc 𝒫 1o ∈ V
27	26	elsn 3721	. . 3 ⊢ (suc 𝒫 1o ∈ {2o} ↔ suc 𝒫 1o = 2o)
28	25, 27	mtbir 682	. 2 ⊢ ¬ suc 𝒫 1o ∈ {2o}
29		ioran 764	. . 3 ⊢ (¬ (suc 𝒫 1o ∈ 2o ∨ suc 𝒫 1o ∈ {2o}) ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
30		df-3o 6679	. . . . . 6 ⊢ 3o = suc 2o
31		df-suc 4511	. . . . . 6 ⊢ suc 2o = (2o ∪ {2o})
32	30, 31	eqtri 2259	. . . . 5 ⊢ 3o = (2o ∪ {2o})
33	32	eleq2i 2305	. . . 4 ⊢ (suc 𝒫 1o ∈ 3o ↔ suc 𝒫 1o ∈ (2o ∪ {2o}))
34		elun 3370	. . . 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 692	. 2 ⊢ (¬ suc 𝒫 1o ∈ 3o ↔ (¬ suc 𝒫 1o ∈ 2o ∧ ¬ suc 𝒫 1o ∈ {2o}))
37	18, 28, 36	mpbir2an 955	1 ⊢ ¬ suc 𝒫 1o ∈ 3o

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ∪ cun 3218  ∅c0 3520  𝒫 cpw 3685  {csn 3705  {cpr 3706  suc csuc 4505  1_oc1o 6670  2_oc2o 6671  3_oc3o 6672

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677  df-2o 6678  df-3o 6679

This theorem is referenced by:  onntri35  7586