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Theorem omnord1 43208
Description: When the same non-zero ordinal is multiplied on the right, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 4-Feb-2025.)
Assertion
Ref Expression
omnord1 𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))
Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem omnord1
StepHypRef Expression
1 omnord1ex 43207 . 2 ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2 1on 8530 . . 3 1o ∈ On
3 2on 8532 . . . 4 2o ∈ On
4 omelon 9711 . . . . . 6 ω ∈ On
5 peano1 7923 . . . . . 6 ∅ ∈ ω
6 ondif1 8553 . . . . . 6 (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
74, 5, 6mpbir2an 710 . . . . 5 ω ∈ (On ∖ 1o)
8 oveq2 7453 . . . . . . . . 9 (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9 oveq2 7453 . . . . . . . . 9 (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
108, 9eleq12d 2832 . . . . . . . 8 (𝑐 = ω → ((1o ·o 𝑐) ∈ (2o ·o 𝑐) ↔ (1o ·o ω) ∈ (2o ·o ω)))
1110bibi2d 342 . . . . . . 7 (𝑐 = ω → ((1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
1211notbid 318 . . . . . 6 (𝑐 = ω → (¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
1312rspcev 3631 . . . . 5 ((ω ∈ (On ∖ 1o) ∧ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
147, 13mpan 689 . . . 4 (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
15 eleq2 2827 . . . . . . . 8 (𝑏 = 2o → (1o𝑏 ↔ 1o ∈ 2o))
16 oveq1 7452 . . . . . . . . 9 (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
1716eleq2d 2824 . . . . . . . 8 (𝑏 = 2o → ((1o ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
1815, 17bibi12d 345 . . . . . . 7 (𝑏 = 2o → ((1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
1918notbid 318 . . . . . 6 (𝑏 = 2o → (¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
2019rexbidv 3181 . . . . 5 (𝑏 = 2o → (∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
2120rspcev 3631 . . . 4 ((2o ∈ On ∧ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
223, 14, 21sylancr 586 . . 3 (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
23 eleq1 2826 . . . . . . . 8 (𝑎 = 1o → (𝑎𝑏 ↔ 1o𝑏))
24 oveq1 7452 . . . . . . . . 9 (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
2524eleq1d 2823 . . . . . . . 8 (𝑎 = 1o → ((𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
2623, 25bibi12d 345 . . . . . . 7 (𝑎 = 1o → ((𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
2726notbid 318 . . . . . 6 (𝑎 = 1o → (¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
2827rexbidv 3181 . . . . 5 (𝑎 = 1o → (∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
2928rexbidv 3181 . . . 4 (𝑎 = 1o → (∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
3029rspcev 3631 . . 3 ((1o ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
312, 22, 30sylancr 586 . 2 (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
321, 31ax-mp 5 1 𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2103  wrex 3072  cdif 3967  c0 4347  Oncon0 6394  (class class class)co 7445  ωcom 7899  1oc1o 8511  2oc2o 8512   ·o comu 8516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766  ax-inf2 9706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-oadd 8522  df-omul 8523
This theorem is referenced by: (None)
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