Theorem tz7.5 2964

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 2938	. 2 |- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))
2		ordwe 2956	. 2 |- (Ord A -> E We A)
3	1, 2	syl3an1 858	1 |- ((Ord A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ w3a 774   = wceq 954   =/= wne 1582  E.wrex 1643   i^i cin 2042   (_ wss 2043  (/)c0 2276  Ecep 2825   We wwe 2911  Ord word 2942

This theorem is referenced by:  tz7.7 2968  onint 3001  tfi 3121  peano5 3148

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-op 2412  df-br 2615  df-opab 2662  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946

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