Theorem oaordnr 43314

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 43313	. 2 ⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
2		0elon 6437	. . 3 ⊢ ∅ ∈ On
3		1on 8519	. . . 4 ⊢ 1o ∈ On
4		omelon 9687	. . . . 5 ⊢ ω ∈ On
5		oveq2 7440	. . . . . . . . 9 ⊢ (𝑐 = ω → (∅ +o 𝑐) = (∅ +o ω))
6		oveq2 7440	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o +o 𝑐) = (1o +o ω))
7	5, 6	eleq12d 2834	. . . . . . . 8 ⊢ (𝑐 = ω → ((∅ +o 𝑐) ∈ (1o +o 𝑐) ↔ (∅ +o ω) ∈ (1o +o ω)))
8	7	bibi2d 342	. . . . . . 7 ⊢ (𝑐 = ω → ((∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
9	8	notbid 318	. . . . . 6 ⊢ (𝑐 = ω → (¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
10	9	rspcev 3621	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
11	4, 10	mpan 690	. . . 4 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
12		eleq2 2829	. . . . . . . 8 ⊢ (𝑏 = 1o → (∅ ∈ 𝑏 ↔ ∅ ∈ 1o))
13		oveq1 7439	. . . . . . . . 9 ⊢ (𝑏 = 1o → (𝑏 +o 𝑐) = (1o +o 𝑐))
14	13	eleq2d 2826	. . . . . . . 8 ⊢ (𝑏 = 1o → ((∅ +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
15	12, 14	bibi12d 345	. . . . . . 7 ⊢ (𝑏 = 1o → ((∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
16	15	notbid 318	. . . . . 6 ⊢ (𝑏 = 1o → (¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
17	16	rexbidv 3178	. . . . 5 ⊢ (𝑏 = 1o → (∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
18	17	rspcev 3621	. . . 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 587	. . 3 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
20		eleq1 2828	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑏 ↔ ∅ ∈ 𝑏))
21		oveq1 7439	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 +o 𝑐) = (∅ +o 𝑐))
22	21	eleq1d 2825	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
23	20, 22	bibi12d 345	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
24	23	notbid 318	. . . . . 6 ⊢ (𝑎 = ∅ → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
25	24	rexbidv 3178	. . . . 5 ⊢ (𝑎 = ∅ → (∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
26	25	rexbidv 3178	. . . 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 3621	. . 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 587	. 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 206   = wceq 1539   ∈ wcel 2107  ∃wrex 3069  ∅c0 4332  Oncon0 6383  (class class class)co 7432  ωcom 7888  1_oc1o 8500   +_o coa 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511

This theorem is referenced by: (None)