Theorem oenord1 43340

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 43339	. 2 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
2		2on 8494	. . . 4 ⊢ 2o ∈ On
3		1oex 8490	. . . . . 6 ⊢ 1o ∈ V
4	3	prid2 4739	. . . . 5 ⊢ 1o ∈ {∅, 1o}
5		df2o3 8488	. . . . 5 ⊢ 2o = {∅, 1o}
6	4, 5	eleqtrri 2833	. . . 4 ⊢ 1o ∈ 2o
7		ondif2 8514	. . . 4 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
8	2, 6, 7	mpbir2an 711	. . 3 ⊢ 2o ∈ (On ∖ 2o)
9		3on 8498	. . . . 5 ⊢ 3o ∈ On
10	3	tpid2 4746	. . . . . 6 ⊢ 1o ∈ {∅, 1o, 2o}
11		df3o2 43337	. . . . . 6 ⊢ 3o = {∅, 1o, 2o}
12	10, 11	eleqtrri 2833	. . . . 5 ⊢ 1o ∈ 3o
13		ondif2 8514	. . . . 5 ⊢ (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
14	9, 12, 13	mpbir2an 711	. . . 4 ⊢ 3o ∈ (On ∖ 2o)
15		omelon 9660	. . . . . 6 ⊢ ω ∈ On
16		peano1 7884	. . . . . 6 ⊢ ∅ ∈ ω
17		ondif1 8513	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
18	15, 16, 17	mpbir2an 711	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
19		oveq2 7413	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ↑o 𝑐) = (2o ↑o ω))
20		oveq2 7413	. . . . . . . . 9 ⊢ (𝑐 = ω → (3o ↑o 𝑐) = (3o ↑o ω))
21	19, 20	eleq12d 2828	. . . . . . . 8 ⊢ (𝑐 = ω → ((2o ↑o 𝑐) ∈ (3o ↑o 𝑐) ↔ (2o ↑o ω) ∈ (3o ↑o ω)))
22	21	bibi2d 342	. . . . . . 7 ⊢ (𝑐 = ω → ((2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
23	22	notbid 318	. . . . . 6 ⊢ (𝑐 = ω → (¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
24	23	rspcev 3601	. . . . 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 690	. . . 4 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
26		eleq2 2823	. . . . . . . 8 ⊢ (𝑏 = 3o → (2o ∈ 𝑏 ↔ 2o ∈ 3o))
27		oveq1 7412	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝑏 ↑o 𝑐) = (3o ↑o 𝑐))
28	27	eleq2d 2820	. . . . . . . 8 ⊢ (𝑏 = 3o → ((2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
29	26, 28	bibi12d 345	. . . . . . 7 ⊢ (𝑏 = 3o → ((2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
30	29	notbid 318	. . . . . 6 ⊢ (𝑏 = 3o → (¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
31	30	rexbidv 3164	. . . . 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 3601	. . . 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 587	. . 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 2822	. . . . . . 7 ⊢ (𝑎 = 2o → (𝑎 ∈ 𝑏 ↔ 2o ∈ 𝑏))
35		oveq1 7412	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑐) = (2o ↑o 𝑐))
36	35	eleq1d 2819	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
37	34, 36	bibi12d 345	. . . . . 6 ⊢ (𝑎 = 2o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
38	37	notbid 318	. . . . 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 3601	. . 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 587	. 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 206   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   ∖ cdif 3923  ∅c0 4308  {cpr 4603  {ctp 4605  Oncon0 6352  (class class class)co 7405  ωcom 7861  1_oc1o 8473  2_oc2o 8474  3_oc3o 8475   ↑_o coe 8479

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729  ax-inf2 9655

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-3o 8482  df-oadd 8484  df-omul 8485  df-oexp 8486

This theorem is referenced by: (None)