Theorem omnord1 44133

Description: When the same nonzero 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 44132	. 2 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2		1on 8471	. . 3 ⊢ 1o ∈ On
3		2on 8472	. . . 4 ⊢ 2o ∈ On
4		omelon 9628	. . . . . 6 ⊢ ω ∈ On
5		peano1 7888	. . . . . 6 ⊢ ∅ ∈ ω
6		ondif1 8491	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
7	4, 5, 6	mpbir2an 724	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
8		oveq2 7424	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9		oveq2 7424	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
10	8, 9	eleq12d 2856	. . . . . . . 8 ⊢ (𝑐 = ω → ((1o ·o 𝑐) ∈ (2o ·o 𝑐) ↔ (1o ·o ω) ∈ (2o ·o ω)))
11	10	bibi2d 345	. . . . . . 7 ⊢ (𝑐 = ω → ((1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
12	11	notbid 321	. . . . . 6 ⊢ (𝑐 = ω → (¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))))
13	12	rspcev 3579	. . . . 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 703	. . . 4 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
15		eleq2 2851	. . . . . . . 8 ⊢ (𝑏 = 2o → (1o ∈ 𝑏 ↔ 1o ∈ 2o))
16		oveq1 7423	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
17	16	eleq2d 2848	. . . . . . . 8 ⊢ (𝑏 = 2o → ((1o ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
18	15, 17	bibi12d 348	. . . . . . 7 ⊢ (𝑏 = 2o → ((1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
19	18	notbid 321	. . . . . 6 ⊢ (𝑏 = 2o → (¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐))))
20	19	rexbidv 3188	. . . . 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 3579	. . . 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 599	. . 3 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
23		eleq1 2850	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 ∈ 𝑏 ↔ 1o ∈ 𝑏))
24		oveq1 7423	. . . . . . . . 9 ⊢ (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
25	24	eleq1d 2847	. . . . . . . 8 ⊢ (𝑎 = 1o → ((𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐) ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
26	23, 25	bibi12d 348	. . . . . . 7 ⊢ (𝑎 = 1o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
27	26	notbid 321	. . . . . 6 ⊢ (𝑎 = 1o → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐)) ↔ ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐))))
28	27	rexbidv 3188	. . . . 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 3188	. . . 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 3579	. . 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 599	. 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))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ∖ cdif 3899  ∅c0 4282  Oncon0 6361  (class class class)co 7416  ωcom 7865  1_oc1o 8451  2_oc2o 8452   ·_o comu 8456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-omul 8463

This theorem is used by: (None)