Theorem onntri3or 7554

Description: Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)

Assertion

Ref	Expression
onntri3or	⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Distinct variable group:   𝑥,𝑦

Proof of Theorem onntri3or

Step	Hyp	Ref	Expression
1		onntri51 7549	. . 3 ⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2		onntri13 7547	. . 3 ⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
3	1, 2	syl 14	. 2 ⊢ (¬ ¬ EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
4		onntri35 7546	. 2 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)
5	3, 4	impbii 126	1 ⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ w3o 1004  ∀wral 2520  EXMIDwem 4306  Oncon0 4483

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-int 3949  df-tr 4208  df-exmid 4307  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-1o 6646  df-2o 6647  df-3o 6648

This theorem is referenced by: (None)