Theorem oenord1ex 43308

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 8399	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4725	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43306	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2827	. . 3 ⊢ 2o ∈ 3o
5		ordom 7809	. . . 4 ⊢ Ord ω
6		ordirr 6325	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8560	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8398	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4715	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8396	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2827	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43305	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 692	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8562	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4722	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2827	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43305	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 692	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2821	. . . . 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 1540   ∈ wcel 2109  ∅c0 4284  {cpr 4579  {ctp 4581  Ord word 6306  (class class class)co 7349  ωcom 7799  1_oc1o 8381  2_oc2o 8382  3_oc3o 8383   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-3o 8390  df-oadd 8392  df-omul 8393  df-oexp 8394

This theorem is referenced by:  oenord1  43309