Theorem oenord1ex 43407

Description: When ordinals two and three are both raised to the power of omega, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oenord1ex	⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))

Proof of Theorem oenord1ex

Step	Hyp	Ref	Expression
1		2oex 8396	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4723	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43405	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2830	. . 3 ⊢ 2o ∈ 3o
5		ordom 7806	. . . 4 ⊢ Ord ω
6		ordirr 6324	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8557	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8395	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4713	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8393	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2830	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43404	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 692	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8559	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4720	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2830	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43404	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 692	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2824	. . . . 5 ⊢ ((2o ↑o ω) ∈ (3o ↑o ω) ↔ ω ∈ ω)
20	6, 19	sylnibr 329	. . . 4 ⊢ (Ord ω → ¬ (2o ↑o ω) ∈ (3o ↑o ω))
21	5, 20	ax-mp 5	. . 3 ⊢ ¬ (2o ↑o ω) ∈ (3o ↑o ω)
22	4, 21	2th 264	. 2 ⊢ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω))
23		xor3 382	. 2 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) ↔ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω)))
24	22, 23	mpbir 231	1 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∅c0 4280  {cpr 4575  {ctp 4577  Ord word 6305  (class class class)co 7346  ωcom 7796  1_oc1o 8378  2_oc2o 8379  3_oc3o 8380   ↑_o coe 8384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-3o 8387  df-oadd 8389  df-omul 8390  df-oexp 8391

This theorem is referenced by:  oenord1  43408