Theorem oawordriexmid 6484

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 6483. (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 6437	. . . . 5 ⊢ 1o ∈ On
2		oawordriexmid.1	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
3	2	3expa 1204	. . . . . . 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 2540	. . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
6		oveq2 5896	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑎 +o 𝑐) = (𝑎 +o 1o))
7		oveq2 5896	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑏 +o 𝑐) = (𝑏 +o 1o))
8	6, 7	sseq12d 3198	. . . . . . . 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 2849	. . . . 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 6481	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +o 1o) = suc 𝑎)
14	13	adantr 276	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 1o) = suc 𝑎)
15		oa1suc 6481	. . . . . 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 3198	. . . 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 2541	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏)
20	19	onsucsssucexmid 4538	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 979   = wceq 1363   ∈ wcel 2158  ∀wral 2465   ⊆ wss 3141  Oncon0 4375  suc csuc 4377  (class class class)co 5888  1_oc1o 6423   +_o coa 6427

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434

This theorem is referenced by: (None)