Theorem inton 4378

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 4377	. 2 ⊢ ∅ ∈ On
2		int0el 3861	. 2 ⊢ (∅ ∈ On → ∩ On = ∅)
3	1, 2	ax-mp 5	1 ⊢ ∩ On = ∅

Syntax hints:   = wceq 1348   ∈ wcel 2141  ∅c0 3414  ∩ cint 3831  Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by: (None)