Theorem onint0 3002

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 3001	. . . . 5 |- ((A (_ On /\ A =/= (/)) -> |^|A e. A)
2		0ex 2706	. . . . . . 7 |- (/) e. V
3		eleq1 1531	. . . . . . 7 |- (|^|A = (/) -> (|^|A e. V <-> (/) e. V))
4	2, 3	mpbiri 194	. . . . . 6 |- (|^|A = (/) -> |^|A e. V)
5		intex 2724	. . . . . 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 1531	. . . . 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 2556	. 2 |- ((/) e. A -> |^|A = (/))
13	11, 12	impbid1 516	1 |- (A (_ On -> (|^|A = (/) <-> (/) e. A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582  Vcvv 1807   (_ wss 2043  (/)c0 2276  |^|cint 2528  Oncon0 2943

This theorem is referenced by:  rankeq0 4676  cfeq0 4894

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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