Theorem onint0 2997

Description: The intersection of a class of ordinal numbers is zero iff the class contains zero.

Assertion

Ref	Expression
onint0	|- (A (_ On -> (|^|A = (/) <-> (/) e. A))

Proof of Theorem onint0

Step	Hyp	Ref	Expression
1		onint 2996	. . . . 5 |- ((A (_ On /\ A =/= (/)) -> |^|A e. A)
2		0ex 2701	. . . . . . 7 |- (/) e. V
3		eleq1 1526	. . . . . . 7 |- (|^|A = (/) -> (|^|A e. V <-> (/) e. V))
4	2, 3	mpbiri 194	. . . . . 6 |- (|^|A = (/) -> |^|A e. V)
5		intex 2719	. . . . . 6 |- (A =/= (/) <-> |^|A e. V)
6	4, 5	sylibr 200	. . . . 5 |- (|^|A = (/) -> A =/= (/))
7	1, 6	sylan2 451	. . . 4 |- ((A (_ On /\ |^|A = (/)) -> |^|A e. A)
8		eleq1 1526	. . . . 5 |- (|^|A = (/) -> (|^|A e. A <-> (/) e. A))
9	8	adantl 388	. . . 4 |- ((A (_ On /\ |^|A = (/)) -> (|^|A e. A <-> (/) e. A))
10	7, 9	mpbid 195	. . 3 |- ((A (_ On /\ |^|A = (/)) -> (/) e. A)
11	10	ex 373	. 2 |- (A (_ On -> (|^|A = (/) -> (/) e. A))
12		int0el 2551	. 2 |- ((/) e. A -> |^|A = (/))
13	11, 12	impbid1 515	1 |- (A (_ On -> (|^|A = (/) <-> (/) e. A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802   (_ wss 2037  (/)c0 2270  |^|cint 2523  Oncon0 2938

This theorem is referenced by:  rankeq0 4668  cfeq0 4886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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