Theorem oenord1ex 43743

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 8416	. . . . 5 ⊢ 2o ∈ V
2	1	tpid3 4717	. . . 4 ⊢ 2o ∈ {∅, 1o, 2o}
3		df3o2 43741	. . . 4 ⊢ 3o = {∅, 1o, 2o}
4	2, 3	eleqtrri 2835	. . 3 ⊢ 2o ∈ 3o
5		ordom 7827	. . . 4 ⊢ Ord ω
6		ordirr 6341	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		2onn 8578	. . . . . . 7 ⊢ 2o ∈ ω
8		1oex 8415	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	prid2 4707	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
10		df2o3 8413	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	9, 10	eleqtrri 2835	. . . . . . 7 ⊢ 1o ∈ 2o
12		nnoeomeqom 43740	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
13	7, 11, 12	mp2an 693	. . . . . 6 ⊢ (2o ↑o ω) = ω
14		3onn 8580	. . . . . . 7 ⊢ 3o ∈ ω
15	8	tpid2 4714	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o}
16	15, 3	eleqtrri 2835	. . . . . . 7 ⊢ 1o ∈ 3o
17		nnoeomeqom 43740	. . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω)
18	14, 16, 17	mp2an 693	. . . . . 6 ⊢ (3o ↑o ω) = ω
19	13, 18	eleq12i 2829	. . . . 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 4273  {cpr 4569  {ctp 4571  Ord word 6322  (class class class)co 7367  ωcom 7817  1_oc1o 8398  2_oc2o 8399  3_oc3o 8400   ↑_o coe 8404

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-3o 8407  df-oadd 8409  df-omul 8410  df-oexp 8411

This theorem is referenced by:  oenord1  43744