Theorem oenord1ex 42520

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 8472	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4769	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 42518	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2824	. . 3 ⊢ 2o ∈ 3o
5		ordom 7858	. . . 4 ⊢ Ord ω
6		ordirr 6372	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8636	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8471	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4759	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8469	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2824	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 42517	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 689	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8638	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4766	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2824	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 42517	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 689	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2818	. . . . 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 1533   ∈ wcel 2098  ∅c0 4314  {cpr 4622  {ctp 4624  Ord word 6353  (class class class)co 7401  ωcom 7848  1_oc1o 8454  2_oc2o 8455  3_oc3o 8456   ↑_o coe 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-inf2 9631

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-3o 8463  df-oadd 8465  df-omul 8466  df-oexp 8467

This theorem is referenced by:  oenord1  42521