Theorem oenord1ex 43305

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 8516	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4778	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43303	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2838	. . 3 ⊢ 2o ∈ 3o
5		ordom 7897	. . . 4 ⊢ Ord ω
6		ordirr 6404	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8679	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8515	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4768	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8513	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2838	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43302	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 692	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8681	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4775	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2838	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43302	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 692	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2832	. . . . 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 2106  ∅c0 4339  {cpr 4633  {ctp 4635  Ord word 6385  (class class class)co 7431  ωcom 7887  1_oc1o 8498  2_oc2o 8499  3_oc3o 8500   ↑_o coe 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-3o 8507  df-oadd 8509  df-omul 8510  df-oexp 8511

This theorem is referenced by:  oenord1  43306