Theorem oawordriexmid 6271

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 6270. (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 6226	. . . . 5 ⊢ 1o ∈ On
2		oawordriexmid.1	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
3	2	3expa 1146	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
4	3	expcom 115	. . . . . 6 ⊢ (𝑐 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))))
5	4	rgen 2439	. . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
6		oveq2 5698	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑎 +o 𝑐) = (𝑎 +o 1o))
7		oveq2 5698	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑏 +o 𝑐) = (𝑏 +o 1o))
8	6, 7	sseq12d 3070	. . . . . . . 8 ⊢ (𝑐 = 1o → ((𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐) ↔ (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))
9	8	imbi2d 229	. . . . . . 7 ⊢ (𝑐 = 1o → ((𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)) ↔ (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o))))
10	9	imbi2d 229	. . . . . 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 2732	. . . . 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 6268	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +o 1o) = suc 𝑎)
14	13	adantr 271	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 1o) = suc 𝑎)
15		oa1suc 6268	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑏 +o 1o) = suc 𝑏)
16	15	adantl 272	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +o 1o) = suc 𝑏)
17	14, 16	sseq12d 3070	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑎 +o 1o) ⊆ (𝑏 +o 1o) ↔ suc 𝑎 ⊆ suc 𝑏))
18	12, 17	sylibd 148	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏))
19	18	rgen2a 2440	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏)
20	19	onsucsssucexmid 4371	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 667   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  ∀wral 2370   ⊆ wss 3013  Oncon0 4214  suc csuc 4216  (class class class)co 5690  1_oc1o 6212   +_o coa 6216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223

This theorem is referenced by: (None)