Theorem omnord1 44307

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 44306	. 2 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2		1on 8489	. . 3 ⊢ 1o ∈ On
3		2on 8490	. . . 4 ⊢ 2o ∈ On
4		omelon 9647	. . . . . 6 ⊢ ω ∈ On
5		peano1 7900	. . . . . 6 ⊢ ∅ ∈ ω
6		ondif1 8509	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
7	4, 5, 6	mpbir2an 724	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
8		oveq2 7428	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9		oveq2 7428	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
10	8, 9	eleq12d 2854	. . . . . . . 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 3576	. . . . 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 2849	. . . . . . . 8 ⊢ (𝑏 = 2o → (1o ∈ 𝑏 ↔ 1o ∈ 2o))
16		oveq1 7427	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
17	16	eleq2d 2846	. . . . . . . 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 3186	. . . . 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 3576	. . . 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 2848	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 ∈ 𝑏 ↔ 1o ∈ 𝑏))
24		oveq1 7427	. . . . . . . . 9 ⊢ (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
25	24	eleq1d 2845	. . . . . . . 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 3186	. . . . 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 3186	. . . 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 3576	. . 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 1569   ∈ wcel 2144  ∃wrex 3086   ∖ cdif 4045  ∅c0 4278  Oncon0 6362  (class class class)co 7420  ωcom 7877  1_oc1o 8469  2_oc2o 8470   ·_o comu 8474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481

This theorem is used by: (None)