Theorem omnord1 43269

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) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem omnord1

Step	Hyp	Ref	Expression
1		omnord1ex 43268	. 2 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2		1on 8536	. . 3 ⊢ 1o ∈ On
3		2on 8538	. . . 4 ⊢ 2o ∈ On
4		omelon 9717	. . . . . 6 ⊢ ω ∈ On
5		peano1 7929	. . . . . 6 ⊢ ∅ ∈ ω
6		ondif1 8559	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
7	4, 5, 6	mpbir2an 710	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
8		oveq2 7458	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9		oveq2 7458	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
10	8, 9	eleq12d 2838	. . . . . . . 8 ⊢ (𝑐 = ω → ((1o ·o 𝑐) ∈ (2o ·o 𝑐) ↔ (1o ·o ω) ∈ (2o ·o ω)))
11	10	bibi2d 342	. . . . . . 7 ⊢ (𝑐 = ω → ((1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
12	11	notbid 318	. . . . . 6 ⊢ (𝑐 = ω → (¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
13	12	rspcev 3635	. . . . 5 ⊢ ((ω ∈ (On ∖ 1o) ∧ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
14	7, 13	mpan 689	. . . 4 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
15		eleq2 2833	. . . . . . . 8 ⊢ (𝑏 = 2o → (1o ∈ 𝑏 ↔ 1o ∈ 2o))
16		oveq1 7457	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
17	16	eleq2d 2830	. . . . . . . 8 ⊢ (𝑏 = 2o → ((1o ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
18	15, 17	bibi12d 345	. . . . . . 7 ⊢ (𝑏 = 2o → ((1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
19	18	notbid 318	. . . . . 6 ⊢ (𝑏 = 2o → (¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
20	19	rexbidv 3185	. . . . 5 ⊢ (𝑏 = 2o → (∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
21	20	rspcev 3635	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
22	3, 14, 21	sylancr 586	. . 3 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
23		eleq1 2832	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 ∈ 𝑏 ↔ 1o ∈ 𝑏))
24		oveq1 7457	. . . . . . . . 9 ⊢ (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
25	24	eleq1d 2829	. . . . . . . 8 ⊢ (𝑎 = 1o → ((𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
26	23, 25	bibi12d 345	. . . . . . 7 ⊢ (𝑎 = 1o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
27	26	notbid 318	. . . . . 6 ⊢ (𝑎 = 1o → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
28	27	rexbidv 3185	. . . . 5 ⊢ (𝑎 = 1o → (∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
29	28	rexbidv 3185	. . . 4 ⊢ (𝑎 = 1o → (∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
30	29	rspcev 3635	. . 3 ⊢ ((1o ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
31	2, 22, 30	sylancr 586	. 2 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
32	1, 31	ax-mp 5	1 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ∖ cdif 3973  ∅c0 4352  Oncon0 6397  (class class class)co 7450  ωcom 7905  1_oc1o 8517  2_oc2o 8518   ·_o comu 8522

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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772  ax-inf2 9712

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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-oadd 8528  df-omul 8529

This theorem is referenced by: (None)