Theorem oaordnr 44444

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 44443	. 2 ⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
2		0elon 6425	. . 3 ⊢ ∅ ∈ On
3		1on 8503	. . . 4 ⊢ 1o ∈ On
4		omelon 9660	. . . . 5 ⊢ ω ∈ On
5		oveq2 7438	. . . . . . . . 9 ⊢ (𝑐 = ω → (∅ +o 𝑐) = (∅ +o ω))
6		oveq2 7438	. . . . . . . . 9 ⊢ (𝑐 = ω → (1o +o 𝑐) = (1o +o ω))
7	5, 6	eleq12d 2858	. . . . . . . 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 3580	. . . . 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 2853	. . . . . . . 8 ⊢ (𝑏 = 1o → (∅ ∈ 𝑏 ↔ ∅ ∈ 1o))
13		oveq1 7437	. . . . . . . . 9 ⊢ (𝑏 = 1o → (𝑏 +o 𝑐) = (1o +o 𝑐))
14	13	eleq2d 2850	. . . . . . . 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 3190	. . . . 5 ⊢ (𝑏 = 1o → (∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 1o ↔ (∅ +o 𝑐) ∈ (1o +o 𝑐))))
18	17	rspcev 3580	. . . 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 2852	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑏 ↔ ∅ ∈ 𝑏))
21		oveq1 7437	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 +o 𝑐) = (∅ +o 𝑐))
22	21	eleq1d 2849	. . . . . . . 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 3190	. . . . 5 ⊢ (𝑎 = ∅ → (∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) ↔ ∃𝑐 ∈ On ¬ (∅ ∈ 𝑏 ↔ (∅ +o 𝑐) ∈ (𝑏 +o 𝑐))))
26	25	rexbidv 3190	. . . 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 3580	. . 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 1569   ∈ wcel 2146  ∃wrex 3090  ∅c0 4281  Oncon0 6369  (class class class)co 7430  ωcom 7887  1_oc1o 8483   +_o coa 8487

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761  ax-inf2 9655

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 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-oadd 8494

This theorem is used by: (None)