Theorem oenord1ex 43277

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 8533	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4798	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43275	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2843	. . 3 ⊢ 2o ∈ 3o
5		ordom 7913	. . . 4 ⊢ Ord ω
6		ordirr 6413	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8698	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8532	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4788	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8530	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2843	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43274	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 691	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8700	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4795	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2843	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43274	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 691	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2837	. . . . 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 1537   ∈ wcel 2108  ∅c0 4352  {cpr 4650  {ctp 4652  Ord word 6394  (class class class)co 7448  ωcom 7903  1_oc1o 8515  2_oc2o 8516  3_oc3o 8517   ↑_o coe 8521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-3o 8524  df-oadd 8526  df-omul 8527  df-oexp 8528

This theorem is referenced by:  oenord1  43278