Theorem oaordnr 44100

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 44099	. 2 ⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
2		0elon 6420	. . 3 ⊢ ∅ ∈ On
3		1on 8472	. . . 4 ⊢ 1o ∈ On
4		omelon 9622	. . . . 5 ⊢ ω ∈ On
5		oveq2 7427	. . . . . . . . 9 ⊢ (𝑐 = ω → (∅ +o 𝑐) = (∅ +o ω))
6		oveq2 7427	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o +o 𝑐) = (1o +o ω))
7	5, 6	eleq12d 2859	. . . . . . . 8 ⊢ (𝑐 = ω → ((∅ +o 𝑐) ∈ (1o +o 𝑐) ↔ (∅ +o ω) ∈ (1o +o ω)))
8	7	bibi2d 345	. . . . . . 7 ⊢ (𝑐 = ω → ((∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
9	8	notbid 321	. . . . . 6 ⊢ (𝑐 = ω → (¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))))
10	9	rspcev 3583	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
11	4, 10	mpan 703	. . . 4 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
12		eleq2 2854	. . . . . . . 8 ⊢ (𝑏 = 1o → (∅ ∈ 𝑏 ↔ ∅ ∈ 1o))
13		oveq1 7426	. . . . . . . . 9 ⊢ (𝑏 = 1o → (𝑏 +o 𝑐) = (1o +o 𝑐))
14	13	eleq2d 2851	. . . . . . . 8 ⊢ (𝑏 = 1o → ((∅ +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐)))
15	12, 14	bibi12d 348	. . . . . . 7 ⊢ (𝑏 = 1o → ((∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
16	15	notbid 321	. . . . . 6 ⊢ (𝑏 = 1o → (¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
17	16	rexbidv 3191	. . . . 5 ⊢ (𝑏 = 1o → (∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
18	17	rspcev 3583	. . . 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 599	. . 3 ⊢ (¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
20		eleq1 2853	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑏 ↔ ∅ ∈ 𝑏))
21		oveq1 7426	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 +o 𝑐) = (∅ +o 𝑐))
22	21	eleq1d 2850	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐) ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)))
23	20, 22	bibi12d 348	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
24	23	notbid 321	. . . . . 6 ⊢ (𝑎 = ∅ → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
25	24	rexbidv 3191	. . . . 5 ⊢ (𝑎 = ∅ → (∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
26	25	rexbidv 3191	. . . 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 3583	. . 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 599	. 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))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  ∅c0 4286  Oncon0 6364  (class class class)co 7419  ωcom 7868  1_oc1o 8452   +_o coa 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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742  ax-inf2 9617

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463

This theorem is used by: (None)