Theorem oaordnr 43741

Description: When the same ordinal is added on the right, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
oaordnr	⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐))

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oaordnr

Step	Hyp	Ref	Expression
1		oaordnrex 43740	. 2 ⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
2		0elon 6365	. . 3 ⊢ ∅ ∈ On
3		1on 8407	. . . 4 ⊢ 1o ∈ On
4		omelon 9558	. . . . 5 ⊢ ω ∈ On
5		oveq2 7364	. . . . . . . . 9 ⊢ (𝑐 = ω → (∅ +o 𝑐) = (∅ +o ω))
6		oveq2 7364	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o +o 𝑐) = (1o +o ω))
7	5, 6	eleq12d 2833	. . . . . . . 8 ⊢ (𝑐 = ω → ((∅ +o 𝑐) ∈ (1o +o 𝑐) ↔ (∅ +o ω) ∈ (1o +o ω)))
8	7	bibi2d 343	. . . . . . 7 ⊢ (𝑐 = ω → ((∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
9	8	notbid 319	. . . . . 6 ⊢ (𝑐 = ω → (¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
10	9	rspcev 3560	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
11	4, 10	mpan 696	. . . 4 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
12		eleq2 2828	. . . . . . . 8 ⊢ (𝑏 = 1o → (∅ ∈ 𝑏 ↔ ∅ ∈ 1o))
13		oveq1 7363	. . . . . . . . 9 ⊢ (𝑏 = 1o → (𝑏 +o 𝑐) = (1o +o 𝑐))
14	13	eleq2d 2825	. . . . . . . 8 ⊢ (𝑏 = 1o → ((∅ +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
15	12, 14	bibi12d 346	. . . . . . 7 ⊢ (𝑏 = 1o → ((∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
16	15	notbid 319	. . . . . 6 ⊢ (𝑏 = 1o → (¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
17	16	rexbidv 3163	. . . . 5 ⊢ (𝑏 = 1o → (∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
18	17	rspcev 3560	. . . 4 ⊢ ((1o ∈ On ∧ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
19	3, 11, 18	sylancr 593	. . 3 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
20		eleq1 2827	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑏 ↔ ∅ ∈ 𝑏))
21		oveq1 7363	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 +o 𝑐) = (∅ +o 𝑐))
22	21	eleq1d 2824	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
23	20, 22	bibi12d 346	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
24	23	notbid 319	. . . . . 6 ⊢ (𝑎 = ∅ → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
25	24	rexbidv 3163	. . . . 5 ⊢ (𝑎 = ∅ → (∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
26	25	rexbidv 3163	. . . 4 ⊢ (𝑎 = ∅ → (∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
27	26	rspcev 3560	. . 3 ⊢ ((∅ ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)))
28	2, 19, 27	sylancr 593	. 2 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)))
29	1, 28	ax-mp 5	1 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐))

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  ∅c0 4261  Oncon0 6310  (class class class)co 7356  ωcom 7806  1_oc1o 8388   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399

This theorem is referenced by: (None)