Theorem oawordriexmid 6134

Description: A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6133. (Contributed by Jim Kingdon, 15-May-2022.)

Hypothesis

Ref	Expression
oawordriexmid.1	⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))

Assertion

Ref	Expression
oawordriexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable groups:   𝑎,𝑏,𝑐   𝜑,𝑎

Allowed substitution hints:   𝜑(𝑏,𝑐)

Proof of Theorem oawordriexmid

Step	Hyp	Ref	Expression
1		1on 6092	. . . . 5 ⊢ 1𝑜 ∈ On
2		oawordriexmid.1	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))
3	2	3expa 1139	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))
4	3	expcom 114	. . . . . 6 ⊢ (𝑐 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))))
5	4	rgen 2421	. . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))
6		oveq2 5571	. . . . . . . . 9 ⊢ (𝑐 = 1𝑜 → (𝑎 +𝑜 𝑐) = (𝑎 +𝑜 1𝑜))
7		oveq2 5571	. . . . . . . . 9 ⊢ (𝑐 = 1𝑜 → (𝑏 +𝑜 𝑐) = (𝑏 +𝑜 1𝑜))
8	6, 7	sseq12d 3037	. . . . . . . 8 ⊢ (𝑐 = 1𝑜 → ((𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐) ↔ (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))
9	8	imbi2d 228	. . . . . . 7 ⊢ (𝑐 = 1𝑜 → ((𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)) ↔ (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜))))
10	9	imbi2d 228	. . . . . 6 ⊢ (𝑐 = 1𝑜 → (((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))) ↔ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))))
11	10	rspcv 2706	. . . . 5 ⊢ (1𝑜 ∈ On → (∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))) → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))))
12	1, 5, 11	mp2 16	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))
13		oa1suc 6131	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +𝑜 1𝑜) = suc 𝑎)
14	13	adantr 270	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +𝑜 1𝑜) = suc 𝑎)
15		oa1suc 6131	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑏 +𝑜 1𝑜) = suc 𝑏)
16	15	adantl 271	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +𝑜 1𝑜) = suc 𝑏)
17	14, 16	sseq12d 3037	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜) ↔ suc 𝑎 ⊆ suc 𝑏))
18	12, 17	sylibd 147	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏))
19	18	rgen2a 2422	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏)
20	19	onsucsssucexmid 4298	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ∀wral 2353   ⊆ wss 2982  Oncon0 4146  suc csuc 4148  (class class class)co 5563  1_𝑜c1o 6078   +_𝑜 coa 6082

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089

This theorem is referenced by: (None)