Theorem tz7.5 2996

Description: A subclass (possibly proper) of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36.

Assertion

Ref	Expression
tz7.5	|- ((Ord A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Distinct variable groups:   x,A   x,B

Proof of Theorem tz7.5

Step	Hyp	Ref	Expression
1		wefrc 2970	. 2 |- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))
2		ordwe 2988	. 2 |- (Ord A -> E We A)
3	1, 2	syl3an1 865	1 |- ((Ord A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ w3a 781   = wceq 992   =/= wne 1628  E.wrex 1692   i^i cin 2098   (_ wss 2099  (/)c0 2332  Ecep 2908   We wwe 2946  Ord word 2974

This theorem is referenced by:  tz7.7 3001  onint 3152  tfi 3207  peano5 3241

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693  df-opab 2741  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978

Copyright terms: Public domain