Theorem onnmin 3010

Description: No member of a set of ordinal numbers belongs to its minimum.

Assertion

Ref	Expression
onnmin	|- ((A (_ On /\ B e. A) -> -. B e. |^|A)

Proof of Theorem onnmin

Step	Hyp	Ref	Expression
1		intss1 2543	. . 3 |- (B e. A -> |^|A (_ B)
2	1	adantl 388	. 2 |- ((A (_ On /\ B e. A) -> |^|A (_ B)
3		ontri1 2976	. . 3 |- ((|^|A e. On /\ B e. On) -> (|^|A (_ B <-> -. B e. |^|A))
4		oninton 3007	. . . 4 |- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
5		ne0i 2282	. . . 4 |- (B e. A -> A =/= (/))
6	4, 5	sylan2 451	. . 3 |- ((A (_ On /\ B e. A) -> |^|A e. On)
7		ssel2 2060	. . 3 |- ((A (_ On /\ B e. A) -> B e. On)
8	3, 6, 7	sylanc 471	. 2 |- ((A (_ On /\ B e. A) -> (|^|A (_ B <-> -. B e. |^|A))
9	2, 8	mpbid 195	1 |- ((A (_ On /\ B e. A) -> -. B e. |^|A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956   =/= wne 1582   (_ wss 2043  (/)c0 2276  |^|cint 2528  Oncon0 2943

This theorem is referenced by:  onnminsb 3011  oneqmin 3013  onminex 3015  onmindif2 3056  cardmin 4840

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-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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