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Theorem oaordnr 42512
Description: When the same ordinal is added on the right, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
Assertion
Ref Expression
oaordnr 𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐))
Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oaordnr
StepHypRef Expression
1 oaordnrex 42511 . 2 ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
2 0elon 6418 . . 3 ∅ ∈ On
3 1on 8484 . . . 4 1o ∈ On
4 omelon 9647 . . . . 5 ω ∈ On
5 oveq2 7420 . . . . . . . . 9 (𝑐 = ω → (∅ +o 𝑐) = (∅ +o ω))
6 oveq2 7420 . . . . . . . . 9 (𝑐 = ω → (1o +o 𝑐) = (1o +o ω))
75, 6eleq12d 2826 . . . . . . . 8 (𝑐 = ω → ((∅ +o 𝑐) ∈ (1o +o 𝑐) ↔ (∅ +o ω) ∈ (1o +o ω)))
87bibi2d 342 . . . . . . 7 (𝑐 = ω → ((∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
98notbid 318 . . . . . 6 (𝑐 = ω → (¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
109rspcev 3612 . . . . 5 ((ω ∈ On ∧ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
114, 10mpan 687 . . . 4 (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
12 eleq2 2821 . . . . . . . 8 (𝑏 = 1o → (∅ ∈ 𝑏 ↔ ∅ ∈ 1o))
13 oveq1 7419 . . . . . . . . 9 (𝑏 = 1o → (𝑏 +o 𝑐) = (1o +o 𝑐))
1413eleq2d 2818 . . . . . . . 8 (𝑏 = 1o → ((∅ +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
1512, 14bibi12d 345 . . . . . . 7 (𝑏 = 1o → ((∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
1615notbid 318 . . . . . 6 (𝑏 = 1o → (¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
1716rexbidv 3177 . . . . 5 (𝑏 = 1o → (∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
1817rspcev 3612 . . . 4 ((1o ∈ On ∧ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
193, 11, 18sylancr 586 . . 3 (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
20 eleq1 2820 . . . . . . . 8 (𝑎 = ∅ → (𝑎𝑏 ↔ ∅ ∈ 𝑏))
21 oveq1 7419 . . . . . . . . 9 (𝑎 = ∅ → (𝑎 +o 𝑐) = (∅ +o 𝑐))
2221eleq1d 2817 . . . . . . . 8 (𝑎 = ∅ → ((𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
2320, 22bibi12d 345 . . . . . . 7 (𝑎 = ∅ → ((𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
2423notbid 318 . . . . . 6 (𝑎 = ∅ → (¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
2524rexbidv 3177 . . . . 5 (𝑎 = ∅ → (∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
2625rexbidv 3177 . . . 4 (𝑎 = ∅ → (∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
2726rspcev 3612 . . 3 ((∅ ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)))
282, 19, 27sylancr 586 . 2 (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)))
291, 28ax-mp 5 1 𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1540  wcel 2105  wrex 3069  c0 4322  Oncon0 6364  (class class class)co 7412  ωcom 7859  1oc1o 8465   +o coa 8469
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-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-oadd 8476
This theorem is referenced by: (None)
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