Theorem onsseli 6445

Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)

Hypotheses

Ref	Expression
on.1	⊢ 𝐴 ∈ On
on.2	⊢ 𝐵 ∈ On

Assertion

Ref	Expression
onsseli	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem onsseli

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		on.2	. 2 ⊢ 𝐵 ∈ On
3		onsseleq 6364	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  Oncon0 6323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  cardom  9910  tskcard  10704  onnolt  28258  bdayfinbndlem1  28459