Theorem 3nelsucpw1 7346

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

Assertion

Ref	Expression
3nelsucpw1	⊢ ¬ 3o ∈ suc 𝒫 1o

Proof of Theorem 3nelsucpw1

Step	Hyp	Ref	Expression
1		1lt2o 6528	. . . . 5 ⊢ 1o ∈ 2o
2		elelsuc 4456	. . . . 5 ⊢ (1o ∈ 2o → 1o ∈ suc 2o)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1o ∈ suc 2o
4		df-3o 6504	. . . 4 ⊢ 3o = suc 2o
5	3, 4	eleqtrri 2281	. . 3 ⊢ 1o ∈ 3o
6		ssnel 4617	. . 3 ⊢ (3o ⊆ 1o → ¬ 1o ∈ 3o)
7	5, 6	mt2 641	. 2 ⊢ ¬ 3o ⊆ 1o
8		pw1ne3 7342	. . . . . 6 ⊢ 𝒫 1o ≠ 3o
9	8	nesymi 2422	. . . . 5 ⊢ ¬ 3o = 𝒫 1o
10	9	a1i 9	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → ¬ 3o = 𝒫 1o)
11		elsuci 4450	. . . 4 ⊢ (3o ∈ suc 𝒫 1o → (3o ∈ 𝒫 1o ∨ 3o = 𝒫 1o))
12	10, 11	ecased 1362	. . 3 ⊢ (3o ∈ suc 𝒫 1o → 3o ∈ 𝒫 1o)
13	12	elpwid 3627	. 2 ⊢ (3o ∈ suc 𝒫 1o → 3o ⊆ 1o)
14	7, 13	mto 664	1 ⊢ ¬ 3o ∈ suc 𝒫 1o

Syntax hints:  ¬ wn 3   = wceq 1373   ∈ wcel 2176   ⊆ wss 3166  𝒫 cpw 3616  suc csuc 4412  1_oc1o 6495  2_oc2o 6496  3_oc3o 6497

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-1o 6502  df-2o 6503  df-3o 6504

This theorem is referenced by:  onntri35  7349