Theorem oenord1 43306

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 43305	. 2 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
2		2on 8519	. . . 4 ⊢ 2o ∈ On
3		1oex 8515	. . . . . 6 ⊢ 1o ∈ V
4	3	prid2 4768	. . . . 5 ⊢ 1o ∈ {∅, 1o}
5		df2o3 8513	. . . . 5 ⊢ 2o = {∅, 1o}
6	4, 5	eleqtrri 2838	. . . 4 ⊢ 1o ∈ 2o
7		ondif2 8539	. . . 4 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
8	2, 6, 7	mpbir2an 711	. . 3 ⊢ 2o ∈ (On ∖ 2o)
9		3on 8523	. . . . 5 ⊢ 3o ∈ On
10	3	tpid2 4775	. . . . . 6 ⊢ 1o ∈ {∅, 1o, 2o}
11		df3o2 43303	. . . . . 6 ⊢ 3o = {∅, 1o, 2o}
12	10, 11	eleqtrri 2838	. . . . 5 ⊢ 1o ∈ 3o
13		ondif2 8539	. . . . 5 ⊢ (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
14	9, 12, 13	mpbir2an 711	. . . 4 ⊢ 3o ∈ (On ∖ 2o)
15		omelon 9684	. . . . . 6 ⊢ ω ∈ On
16		peano1 7911	. . . . . 6 ⊢ ∅ ∈ ω
17		ondif1 8538	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
18	15, 16, 17	mpbir2an 711	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
19		oveq2 7439	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ↑o 𝑐) = (2o ↑o ω))
20		oveq2 7439	. . . . . . . . 9 ⊢ (𝑐 = ω → (3o ↑o 𝑐) = (3o ↑o ω))
21	19, 20	eleq12d 2833	. . . . . . . 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 3622	. . . . 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 2828	. . . . . . . 8 ⊢ (𝑏 = 3o → (2o ∈ 𝑏 ↔ 2o ∈ 3o))
27		oveq1 7438	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝑏 ↑o 𝑐) = (3o ↑o 𝑐))
28	27	eleq2d 2825	. . . . . . . 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 3177	. . . . 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 3622	. . . 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 2827	. . . . . . 7 ⊢ (𝑎 = 2o → (𝑎 ∈ 𝑏 ↔ 2o ∈ 𝑏))
35		oveq1 7438	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑐) = (2o ↑o 𝑐))
36	35	eleq1d 2824	. . . . . . 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 3220	. . . 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 3622	. . 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 1537   ∈ wcel 2106  ∃wrex 3068   ∖ cdif 3960  ∅c0 4339  {cpr 4633  {ctp 4635  Oncon0 6386  (class class class)co 7431  ωcom 7887  1_oc1o 8498  2_oc2o 8499  3_oc3o 8500   ↑_o coe 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-3o 8507  df-oadd 8509  df-omul 8510  df-oexp 8511

This theorem is referenced by: (None)