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Theorem onntri45 7239
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 7224 . . . . 5 𝒫 1o ∈ On
21onsuci 4515 . . . 4 suc 𝒫 1o ∈ On
3 3on 6427 . . . 4 3o ∈ On
4 sseq1 3178 . . . . . . . 8 (𝑥 = suc 𝒫 1o → (𝑥𝑦 ↔ suc 𝒫 1o𝑦))
5 sseq2 3179 . . . . . . . 8 (𝑥 = suc 𝒫 1o → (𝑦𝑥𝑦 ⊆ suc 𝒫 1o))
64, 5orbi12d 793 . . . . . . 7 (𝑥 = suc 𝒫 1o → ((𝑥𝑦𝑦𝑥) ↔ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
76notbid 667 . . . . . 6 (𝑥 = suc 𝒫 1o → (¬ (𝑥𝑦𝑦𝑥) ↔ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
87notbid 667 . . . . 5 (𝑥 = suc 𝒫 1o → (¬ ¬ (𝑥𝑦𝑦𝑥) ↔ ¬ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o)))
9 sseq2 3179 . . . . . . . 8 (𝑦 = 3o → (suc 𝒫 1o𝑦 ↔ suc 𝒫 1o ⊆ 3o))
10 sseq1 3178 . . . . . . . 8 (𝑦 = 3o → (𝑦 ⊆ suc 𝒫 1o ↔ 3o ⊆ suc 𝒫 1o))
119, 10orbi12d 793 . . . . . . 7 (𝑦 = 3o → ((suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
1211notbid 667 . . . . . 6 (𝑦 = 3o → (¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
1312notbid 667 . . . . 5 (𝑦 = 3o → (¬ ¬ (suc 𝒫 1o𝑦𝑦 ⊆ suc 𝒫 1o) ↔ ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
148, 13rspc2v 2854 . . . 4 ((suc 𝒫 1o ∈ On ∧ 3o ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o)))
152, 3, 14mp2an 426 . . 3 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o))
16 ioran 752 . . 3 (¬ (suc 𝒫 1o ⊆ 3o ∨ 3o ⊆ suc 𝒫 1o) ↔ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
1715, 16sylnib 676 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
18 sucpw1nss3 7233 . . 3 EXMID → ¬ suc 𝒫 1o ⊆ 3o)
19 3nsssucpw1 7234 . . 3 EXMID → ¬ 3o ⊆ suc 𝒫 1o)
2018, 19jca 306 . 2 EXMID → (¬ suc 𝒫 1o ⊆ 3o ∧ ¬ 3o ⊆ suc 𝒫 1o))
2117, 20nsyl 628 1 (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  wral 2455  wss 3129  𝒫 cpw 3575  EXMIDwem 4194  Oncon0 4363  suc csuc 4365  1oc1o 6409  3oc3o 6411
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-1o 6416  df-2o 6417  df-3o 6418
This theorem is referenced by:  onntri2or  7244
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