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Mirrors > Home > ILE Home > Th. List > oawordriexmid | GIF version |
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 6373. (Contributed by Jim Kingdon, 15-May-2022.) |
Ref | Expression |
---|---|
oawordriexmid.1 | ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) |
Ref | Expression |
---|---|
oawordriexmid | ⊢ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 6328 | . . . . 5 ⊢ 1o ∈ On | |
2 | oawordriexmid.1 | . . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) | |
3 | 2 | 3expa 1182 | . . . . . . 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 2488 | . . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) |
6 | oveq2 5790 | . . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑎 +o 𝑐) = (𝑎 +o 1o)) | |
7 | oveq2 5790 | . . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑏 +o 𝑐) = (𝑏 +o 1o)) | |
8 | 6, 7 | sseq12d 3133 | . . . . . . . 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 2789 | . . . . 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 6371 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +o 1o) = suc 𝑎) | |
14 | 13 | adantr 274 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 1o) = suc 𝑎) |
15 | oa1suc 6371 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝑏 +o 1o) = suc 𝑏) | |
16 | 15 | adantl 275 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +o 1o) = suc 𝑏) |
17 | 14, 16 | sseq12d 3133 | . . . 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 2489 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏) |
20 | 19 | onsucsssucexmid 4450 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 Oncon0 4293 suc csuc 4295 (class class class)co 5782 1oc1o 6314 +o coa 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 |
This theorem is referenced by: (None) |
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