Theorem oenord1ex 43767

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 8411	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4718	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43765	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2836	. . 3 ⊢ 2o ∈ 3o
5		ordom 7822	. . . 4 ⊢ Ord ω
6		ordirr 6337	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8573	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8410	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4708	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8408	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2836	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43764	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 693	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8575	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4715	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2836	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43764	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 693	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2830	. . . . 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 1542   ∈ wcel 2114  ∅c0 4274  {cpr 4570  {ctp 4572  Ord word 6318  (class class class)co 7362  ωcom 7812  1_oc1o 8393  2_oc2o 8394  3_oc3o 8395   ↑_o coe 8399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684  ax-inf2 9557

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-3o 8402  df-oadd 8404  df-omul 8405  df-oexp 8406

This theorem is referenced by:  oenord1  43768