Theorem oenord1ex 43775

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 8410	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4708	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43773	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2840	. . 3 ⊢ 2o ∈ 3o
5		ordom 7820	. . . 4 ⊢ Ord ω
6		ordirr 6332	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8572	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8409	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4698	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8407	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2840	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43772	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 699	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8574	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4705	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2840	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43772	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 699	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2834	. . . . 5 ⊢ ((2o ↑o ω) ∈ (3o ↑o ω) ↔ ω ∈ ω)
20	6, 19	sylnibr 331	. . . 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 266	. 2 ⊢ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω))
23		xor3 384	. 2 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) ↔ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω)))
24	22, 23	mpbir 233	1 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∅c0 4264  {cpr 4560  {ctp 4562  Ord word 6313  (class class class)co 7360  ωcom 7810  1_oc1o 8392  2_oc2o 8393  3_oc3o 8394   ↑_o coe 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-3o 8401  df-oadd 8403  df-omul 8404  df-oexp 8405

This theorem is referenced by:  oenord1  43776