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Theorem onntri45 7155
 Description: Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
Assertion
Ref Expression
onntri45 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ EXMID)
Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri45
StepHypRef Expression
1 pw1on 7140 . . . . 5 𝒫 1o ∈ On
21onsuci 4469 . . . 4 suc 𝒫 1o ∈ On
3 3on 6364 . . . 4 3o ∈ On
4 sseq1 3147 . . . . . . . 8 (𝑥 = suc 𝒫 1o → (𝑥𝑦 ↔ suc 𝒫 1o𝑦))
5 sseq2 3148 . . . . . . . 8 (𝑥 = suc 𝒫 1o → (𝑦𝑥𝑦 ⊆ suc 𝒫 1o))
64, 5orbi12d 783 . . . . . . 7 (𝑥 = suc 𝒫 1o → ((𝑥𝑦𝑦𝑥) ↔ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
76notbid 657 . . . . . 6 (𝑥 = suc 𝒫 1o → (¬ (𝑥𝑦𝑦𝑥) ↔ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
87notbid 657 . . . . 5 (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥𝑦𝑦𝑥) ↔ ¬ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
9 sseq2 3148 . . . . . . . 8 (𝑦 = 3o → (suc 𝒫 1o𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10 sseq1 3147 . . . . . . . 8 (𝑦 = 3o → (𝑦 ⊆ suc 𝒫 1o ↔ 3o ⊆ suc 𝒫 1o))
119, 10orbi12d 783 . . . . . . 7 (𝑦 = 3o → ((suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
1211notbid 657 . . . . . 6 (𝑦 = 3o → (¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
1312notbid 657 . . . . 5 (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
148, 13rspc2v 2826 . . . 4 ((suc 𝒫 1o ∈ On ∧ 3o ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
152, 3, 14mp2an 423 . . 3 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o))
16 ioran 742 . . 3 (¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
1715, 16sylnib 666 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
18 sucpw1nss3 7149 . . 3 EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19 3nsssucpw1 7150 . . 3 EXMID → ¬ 3o ⊆ suc 𝒫 1o)
2018, 19jca 304 . 2 EXMID → (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
2117, 20nsyl 618 1 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ EXMID)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 2125  ∀wral 2432   ⊆ wss 3098  𝒫 cpw 3539  EXMIDwem 4150  Oncon0 4318  suc csuc 4320  1oc1o 6346  3oc3o 6348 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-uni 3769  df-int 3804  df-tr 4059  df-exmid 4151  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544  df-1o 6353  df-2o 6354  df-3o 6355 This theorem is referenced by:  onntri2or  7160
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