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Theorem oenord1 42532
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 42531 . 2 ¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))
2 2on 8486 . . . 4 2o ∈ On
3 1oex 8482 . . . . . 6 1o ∈ V
43prid2 4767 . . . . 5 1o ∈ {∅, 1o}
5 df2o3 8480 . . . . 5 2o = {∅, 1o}
64, 5eleqtrri 2831 . . . 4 1o ∈ 2o
7 ondif2 8508 . . . 4 (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
82, 6, 7mpbir2an 708 . . 3 2o ∈ (On ∖ 2o)
9 3on 8490 . . . . 5 3o ∈ On
103tpid2 4774 . . . . . 6 1o ∈ {∅, 1o, 2o}
11 df3o2 42529 . . . . . 6 3o = {∅, 1o, 2o}
1210, 11eleqtrri 2831 . . . . 5 1o ∈ 3o
13 ondif2 8508 . . . . 5 (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
149, 12, 13mpbir2an 708 . . . 4 3o ∈ (On ∖ 2o)
15 omelon 9647 . . . . . 6 ω ∈ On
16 peano1 7883 . . . . . 6 ∅ ∈ ω
17 ondif1 8507 . . . . . 6 (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
1815, 16, 17mpbir2an 708 . . . . 5 ω ∈ (On ∖ 1o)
19 oveq2 7420 . . . . . . . . 9 (𝑐 = ω → (2oo 𝑐) = (2oo ω))
20 oveq2 7420 . . . . . . . . 9 (𝑐 = ω → (3oo 𝑐) = (3oo ω))
2119, 20eleq12d 2826 . . . . . . . 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 3612 . . . . 5 ((ω ∈ (On ∖ 1o) ∧ ¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)))
2518, 24mpan 687 . . . 4 (¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐)))
26 eleq2 2821 . . . . . . . 8 (𝑏 = 3o → (2o𝑏 ↔ 2o ∈ 3o))
27 oveq1 7419 . . . . . . . . 9 (𝑏 = 3o → (𝑏o 𝑐) = (3oo 𝑐))
2827eleq2d 2818 . . . . . . . 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 3177 . . . . 5 (𝑏 = 3o → (∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2oo 𝑐) ∈ (3oo 𝑐))))
3231rspcev 3612 . . . 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 586 . . 3 (¬ (2o ∈ 3o ↔ (2oo ω) ∈ (3oo ω)) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o𝑏 ↔ (2oo 𝑐) ∈ (𝑏o 𝑐)))
34 eleq1 2820 . . . . . . 7 (𝑎 = 2o → (𝑎𝑏 ↔ 2o𝑏))
35 oveq1 7419 . . . . . . . 8 (𝑎 = 2o → (𝑎o 𝑐) = (2oo 𝑐))
3635eleq1d 2817 . . . . . . 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 3218 . . . 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 3612 . . 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 586 . 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 205   = wceq 1540  wcel 2105  wrex 3069  cdif 3945  c0 4322  {cpr 4630  {ctp 4632  Oncon0 6364  (class class class)co 7412  ωcom 7859  1oc1o 8465  2oc2o 8466  3oc3o 8467  o coe 8471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-3o 8474  df-oadd 8476  df-omul 8477  df-oexp 8478
This theorem is referenced by: (None)
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