Theorem onelpss 5802

Description: Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)

Assertion

Ref	Expression
onelpss	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)))

Proof of Theorem onelpss

Step	Hyp	Ref	Expression
1		eloni 5771	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 5771	. 2 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordelssne 5788	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)))
4	1, 2, 3	syl2an 493	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2030   ≠ wne 2823   ⊆ wss 3607  Ord word 5760  Oncon0 5761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765

This theorem is referenced by:  tfindsg  7102  findsg  7135  oancom  8586  cardsdom2  8852  alephord  8936  scutbdaylt  32047