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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) → (𝑎𝑏 → (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
StepHypRef Expression
1 1on 6092 . . . . 5 1𝑜 ∈ On
2 oawordriexmid.1 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))
323expa 1139 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑐 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)))
43expcom 114 . . . . . 6 (𝑐 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))))
54rgen 2421 . . . . 5 𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (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))
86, 7sseq12d 3037 . . . . . . . 8 (𝑐 = 1𝑜 → ((𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐) ↔ (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))
98imbi2d 228 . . . . . . 7 (𝑐 = 1𝑜 → ((𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐)) ↔ (𝑎𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜))))
109imbi2d 228 . . . . . 6 (𝑐 = 1𝑜 → (((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))) ↔ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))))
1110rspcv 2706 . . . . 5 (1𝑜 ∈ On → (∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 𝑐) ⊆ (𝑏 +𝑜 𝑐))) → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))))
121, 5, 11mp2 16 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → (𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜)))
13 oa1suc 6131 . . . . . 6 (𝑎 ∈ On → (𝑎 +𝑜 1𝑜) = suc 𝑎)
1413adantr 270 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +𝑜 1𝑜) = suc 𝑎)
15 oa1suc 6131 . . . . . 6 (𝑏 ∈ On → (𝑏 +𝑜 1𝑜) = suc 𝑏)
1615adantl 271 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +𝑜 1𝑜) = suc 𝑏)
1714, 16sseq12d 3037 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑎 +𝑜 1𝑜) ⊆ (𝑏 +𝑜 1𝑜) ↔ suc 𝑎 ⊆ suc 𝑏))
1812, 17sylibd 147 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝑏 → suc 𝑎 ⊆ suc 𝑏))
1918rgen2a 2422 . 2 𝑎 ∈ On ∀𝑏 ∈ On (𝑎𝑏 → suc 𝑎 ⊆ suc 𝑏)
2019onsucsssucexmid 4298 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
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  1𝑜c1o 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)
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