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Theorem oenord1 43329
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) ¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐))
Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenord1
StepHypRef Expression
1 oenord1ex 43328 . 2 ¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))
2 2on 8520 . . . 4 2o ∈ On
3 1oex 8516 . . . . . 6 1o ∈ V
43prid2 4763 . . . . 5 1o ∈ {∅, 1o}
5 df2o3 8514 . . . . 5 2o = {∅, 1o}
64, 5eleqtrri 2840 . . . 4 1o ∈ 2o
7 ondif2 8540 . . . 4 (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
82, 6, 7mpbir2an 711 . . 3 2o ∈ (On ∖ 2o)
9 3on 8524 . . . . 5 3o ∈ On
103tpid2 4770 . . . . . 6 1o ∈ {∅, 1o, 2o}
11 df3o2 43326 . . . . . 6 3o = {∅, 1o, 2o}
1210, 11eleqtrri 2840 . . . . 5 1o ∈ 3o
13 ondif2 8540 . . . . 5 (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
149, 12, 13mpbir2an 711 . . . 4 3o ∈ (On ∖ 2o)
15 omelon 9686 . . . . . 6 ω ∈ On
16 peano1 7910 . . . . . 6 ∅ ∈ ω
17 ondif1 8539 . . . . . 6 (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
1815, 16, 17mpbir2an 711 . . . . 5 ω ∈ (On ∖ 1o)
19 oveq2 7439 . . . . . . . . 9 (𝑐 = ω → (2oo 𝑐) = (2oo ω))
20 oveq2 7439 . . . . . . . . 9 (𝑐 = ω → (3oo 𝑐) = (3oo ω))
2119, 20eleq12d 2835 . . . . . . . 8 (𝑐 = ω → ((2oo 𝑐) ∈ (3oo 𝑐) ↔ (2oo ω) ∈ (3oo ω)))
2221bibi2d 342 . . . . . . 7 (𝑐 = ω → ((2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)) ↔ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))))
2322notbid 318 . . . . . 6 (𝑐 = ω → (¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))))
2423rspcev 3622 . . . . 5 ((ω ∈ (On ∖ 1o) ∧ ¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)))
2518, 24mpan 690 . . . 4 (¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)))
26 eleq2 2830 . . . . . . . 8 (𝑏 = 3o → (2o𝑏 ↔ 2o ∈ 3o))
27 oveq1 7438 . . . . . . . . 9 (𝑏 = 3o → (𝑏o 𝑐) = (3oo 𝑐))
2827eleq2d 2827 . . . . . . . 8 (𝑏 = 3o → ((2oo 𝑐) ∈ (𝑏o 𝑐) ↔ (2oo 𝑐) ∈ (3oo 𝑐)))
2926, 28bibi12d 345 . . . . . . 7 (𝑏 = 3o → ((2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)) ↔ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐))))
3029notbid 318 . . . . . 6 (𝑏 = 3o → (¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐))))
3130rexbidv 3179 . . . . 5 (𝑏 = 3o → (∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐))))
3231rspcev 3622 . . . 4 ((3o ∈ (On ∖ 2o) ∧ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐))) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)))
3314, 25, 32sylancr 587 . . 3 (¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω)) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)))
34 eleq1 2829 . . . . . . 7 (𝑎 = 2o → (𝑎𝑏 ↔ 2o𝑏))
35 oveq1 7438 . . . . . . . 8 (𝑎 = 2o → (𝑎o 𝑐) = (2oo 𝑐))
3635eleq1d 2826 . . . . . . 7 (𝑎 = 2o → ((𝑎o 𝑐) ∈ (𝑏o 𝑐) ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)))
3734, 36bibi12d 345 . . . . . 6 (𝑎 = 2o → ((𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐)) ↔ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐))))
3837notbid 318 . . . . 5 (𝑎 = 2o → (¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐)) ↔ ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐))))
39382rexbidv 3222 . . . 4 (𝑎 = 2o → (∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐)) ↔ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐))))
4039rspcev 3622 . . 3 ((2o ∈ (On ∖ 2o) ∧ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐))) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐)))
418, 33, 40sylancr 587 . 2 (¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω)) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐)))
421, 41ax-mp 5 1 𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎o 𝑐) ∈ (𝑏o 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  c0 4333  {cpr 4628  {ctp 4630  Oncon0 6384  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  3oc3o 8501  o coe 8505
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-3o 8508  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by: (None)
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