Theorem omnord1 43751

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 43750	. 2 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
2		1on 8410	. . 3 ⊢ 1o ∈ On
3		2on 8411	. . . 4 ⊢ 2o ∈ On
4		omelon 9558	. . . . . 6 ⊢ ω ∈ On
5		peano1 7833	. . . . . 6 ⊢ ∅ ∈ ω
6		ondif1 8429	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
7	4, 5, 6	mpbir2an 712	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
8		oveq2 7368	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o ·o 𝑐) = (1o ·o ω))
9		oveq2 7368	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ·o 𝑐) = (2o ·o ω))
10	8, 9	eleq12d 2831	. . . . . . . 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 3565	. . . . 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 691	. . . 4 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 2o ↔ (1o ·o 𝑐) ∈ (2o ·o 𝑐)))
15		eleq2 2826	. . . . . . . 8 ⊢ (𝑏 = 2o → (1o ∈ 𝑏 ↔ 1o ∈ 2o))
16		oveq1 7367	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ·o 𝑐) = (2o ·o 𝑐))
17	16	eleq2d 2823	. . . . . . . 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 3162	. . . . 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 3565	. . . 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 588	. . 3 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (1o ∈ 𝑏 ↔ (1o ·o 𝑐) ∈ (𝑏 ·o 𝑐)))
23		eleq1 2825	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 ∈ 𝑏 ↔ 1o ∈ 𝑏))
24		oveq1 7367	. . . . . . . . 9 ⊢ (𝑎 = 1o → (𝑎 ·o 𝑐) = (1o ·o 𝑐))
25	24	eleq1d 2822	. . . . . . . 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 3162	. . . . 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 3162	. . . 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 3565	. . 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 588	. 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 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3887  ∅c0 4274  Oncon0 6317  (class class class)co 7360  ωcom 7810  1_oc1o 8391  2_oc2o 8392   ·_o comu 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-omul 8403

This theorem is referenced by: (None)