Theorem inton 6378

Description: The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

Assertion

Ref	Expression
inton	⊢ ∩ On = ∅

Proof of Theorem inton

Step	Hyp	Ref	Expression
1		0elon 6374	. 2 ⊢ ∅ ∈ On
2		int0el 4922	. 2 ⊢ (∅ ∈ On → ∩ On = ∅)
3	1, 2	ax-mp 5	1 ⊢ ∩ On = ∅

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ∩ cint 4890  Oncon0 6319

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 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-tr 5194  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-ord 6322  df-on 6323

This theorem is referenced by:  bday0  27821