Theorem oenord1ex 42531

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 8483	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4777	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 42529	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2831	. . 3 ⊢ 2o ∈ 3o
5		ordom 7869	. . . 4 ⊢ Ord ω
6		ordirr 6382	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8647	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8482	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4767	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8480	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2831	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 42528	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 689	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8649	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4774	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2831	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 42528	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 689	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2825	. . . . 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 230	1 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∅c0 4322  {cpr 4630  {ctp 4632  Ord word 6363  (class class class)co 7412  ωcom 7859  1_oc1o 8465  2_oc2o 8466  3_oc3o 8467   ↑_o coe 8471

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-inf2 9642

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-3o 8474  df-oadd 8476  df-omul 8477  df-oexp 8478

This theorem is referenced by:  oenord1  42532