Theorem omnord1 43294

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 43293	. 2 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2		1on 8446	. . 3 ⊢ 1o ∈ On
3		2on 8447	. . . 4 ⊢ 2o ∈ On
4		omelon 9599	. . . . . 6 ⊢ ω ∈ On
5		peano1 7865	. . . . . 6 ⊢ ∅ ∈ ω
6		ondif1 8465	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
7	4, 5, 6	mpbir2an 711	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
8		oveq2 7395	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9		oveq2 7395	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
10	8, 9	eleq12d 2822	. . . . . . . 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 3588	. . . . 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 690	. . . 4 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
15		eleq2 2817	. . . . . . . 8 ⊢ (𝑏 = 2o → (1o ∈ 𝑏 ↔ 1o ∈ 2o))
16		oveq1 7394	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
17	16	eleq2d 2814	. . . . . . . 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 3157	. . . . 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 3588	. . . 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 587	. . 3 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
23		eleq1 2816	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 ∈ 𝑏 ↔ 1o ∈ 𝑏))
24		oveq1 7394	. . . . . . . . 9 ⊢ (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
25	24	eleq1d 2813	. . . . . . . 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 3157	. . . . 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 3157	. . . 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 3588	. . 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 587	. 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 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3911  ∅c0 4296  Oncon0 6332  (class class class)co 7387  ωcom 7842  1_oc1o 8427  2_oc2o 8428   ·_o comu 8432

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-omul 8439

This theorem is referenced by: (None)