Theorem oawordriexmid 6490

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 6489. (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 6443	. . . . 5 ⊢ 1o ∈ On
2		oawordriexmid.1	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
3	2	3expa 1205	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
4	3	expcom 116	. . . . . 6 ⊢ (𝑐 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))))
5	4	rgen 2543	. . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
6		oveq2 5900	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑎 +o 𝑐) = (𝑎 +o 1o))
7		oveq2 5900	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑏 +o 𝑐) = (𝑏 +o 1o))
8	6, 7	sseq12d 3201	. . . . . . . 8 ⊢ (𝑐 = 1o → ((𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐) ↔ (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))
9	8	imbi2d 230	. . . . . . 7 ⊢ (𝑐 = 1o → ((𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)) ↔ (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o))))
10	9	imbi2d 230	. . . . . 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 2852	. . . . 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 6487	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +o 1o) = suc 𝑎)
14	13	adantr 276	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 1o) = suc 𝑎)
15		oa1suc 6487	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑏 +o 1o) = suc 𝑏)
16	15	adantl 277	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +o 1o) = suc 𝑏)
17	14, 16	sseq12d 3201	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑎 +o 1o) ⊆ (𝑏 +o 1o) ↔ suc 𝑎 ⊆ suc 𝑏))
18	12, 17	sylibd 149	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏))
19	18	rgen2a 2544	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏)
20	19	onsucsssucexmid 4541	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∀wral 2468   ⊆ wss 3144  Oncon0 4378  suc csuc 4380  (class class class)co 5892  1_oc1o 6429   +_o coa 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-oadd 6440

This theorem is referenced by: (None)