Theorem oenord1ex 43594

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 8408	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4729	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43592	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2834	. . 3 ⊢ 2o ∈ 3o
5		ordom 7818	. . . 4 ⊢ Ord ω
6		ordirr 6334	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8570	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8407	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4719	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8405	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2834	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43591	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 693	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8572	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4726	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2834	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43591	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 693	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2828	. . . . 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 1542   ∈ wcel 2114  ∅c0 4284  {cpr 4581  {ctp 4583  Ord word 6315  (class class class)co 7358  ωcom 7808  1_oc1o 8390  2_oc2o 8391  3_oc3o 8392   ↑_o coe 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-3o 8399  df-oadd 8401  df-omul 8402  df-oexp 8403

This theorem is referenced by:  oenord1  43595