Theorem oenord1 43774

Description: When two ordinals (both at least as large as two) are raised to the same power, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 4-Feb-2025.)

Assertion

Ref	Expression
oenord1	⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenord1

Step	Hyp	Ref	Expression
1		oenord1ex 43773	. 2 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
2		2on 8412	. . . 4 ⊢ 2o ∈ On
3		1oex 8409	. . . . . 6 ⊢ 1o ∈ V
4	3	prid2 4697	. . . . 5 ⊢ 1o ∈ {∅, 1o}
5		df2o3 8407	. . . . 5 ⊢ 2o = {∅, 1o}
6	4, 5	eleqtrri 2840	. . . 4 ⊢ 1o ∈ 2o
7		ondif2 8431	. . . 4 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
8	2, 6, 7	mpbir2an 718	. . 3 ⊢ 2o ∈ (On ∖ 2o)
9		3on 8415	. . . . 5 ⊢ 3o ∈ On
10	3	tpid2 4704	. . . . . 6 ⊢ 1o ∈ {∅, 1o, 2o}
11		df3o2 43771	. . . . . 6 ⊢ 3o = {∅, 1o, 2o}
12	10, 11	eleqtrri 2840	. . . . 5 ⊢ 1o ∈ 3o
13		ondif2 8431	. . . . 5 ⊢ (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
14	9, 12, 13	mpbir2an 718	. . . 4 ⊢ 3o ∈ (On ∖ 2o)
15		omelon 9562	. . . . . 6 ⊢ ω ∈ On
16		peano1 7832	. . . . . 6 ⊢ ∅ ∈ ω
17		ondif1 8430	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
18	15, 16, 17	mpbir2an 718	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
19		oveq2 7367	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ↑o 𝑐) = (2o ↑o ω))
20		oveq2 7367	. . . . . . . . 9 ⊢ (𝑐 = ω → (3o ↑o 𝑐) = (3o ↑o ω))
21	19, 20	eleq12d 2835	. . . . . . . 8 ⊢ (𝑐 = ω → ((2o ↑o 𝑐) ∈ (3o ↑o 𝑐) ↔ (2o ↑o ω) ∈ (3o ↑o ω)))
22	21	bibi2d 344	. . . . . . 7 ⊢ (𝑐 = ω → ((2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
23	22	notbid 320	. . . . . 6 ⊢ (𝑐 = ω → (¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
24	23	rspcev 3561	. . . . 5 ⊢ ((ω ∈ (On ∖ 1o) ∧ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
25	18, 24	mpan 697	. . . 4 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
26		eleq2 2830	. . . . . . . 8 ⊢ (𝑏 = 3o → (2o ∈ 𝑏 ↔ 2o ∈ 3o))
27		oveq1 7366	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝑏 ↑o 𝑐) = (3o ↑o 𝑐))
28	27	eleq2d 2827	. . . . . . . 8 ⊢ (𝑏 = 3o → ((2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
29	26, 28	bibi12d 347	. . . . . . 7 ⊢ (𝑏 = 3o → ((2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
30	29	notbid 320	. . . . . 6 ⊢ (𝑏 = 3o → (¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
31	30	rexbidv 3165	. . . . 5 ⊢ (𝑏 = 3o → (∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
32	31	rspcev 3561	. . . 4 ⊢ ((3o ∈ (On ∖ 2o) ∧ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
33	14, 25, 32	sylancr 594	. . 3 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
34		eleq1 2829	. . . . . . 7 ⊢ (𝑎 = 2o → (𝑎 ∈ 𝑏 ↔ 2o ∈ 𝑏))
35		oveq1 7366	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑐) = (2o ↑o 𝑐))
36	35	eleq1d 2826	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
37	34, 36	bibi12d 347	. . . . . 6 ⊢ (𝑎 = 2o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
38	37	notbid 320	. . . . 5 ⊢ (𝑎 = 2o → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
39	38	2rexbidv 3206	. . . 4 ⊢ (𝑎 = 2o → (∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
40	39	rspcev 3561	. . 3 ⊢ ((2o ∈ (On ∖ 2o) ∧ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
41	8, 33, 40	sylancr 594	. 2 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
42	1, 41	ax-mp 5	1 ⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   ∖ cdif 3881  ∅c0 4263  {cpr 4559  {ctp 4561  Oncon0 6313  (class class class)co 7359  ωcom 7809  1_oc1o 8392  2_oc2o 8393  3_oc3o 8394   ↑_o coe 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-2o 8400  df-3o 8401  df-oadd 8403  df-omul 8404  df-oexp 8405

This theorem is referenced by: (None)