Theorem tz7.5 2220

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 group(s):   x,A   x,B

Proof of Theorem tz7.5

Step	Hyp	Ref	Expression
1		wefrc 2195	. 2 |- ((E We A /\ (B (_ A /\ -. B = (/))) -> E.x e. B (B i^i x) = (/))
2		ordwe 2212	. 2 |- (Ord A -> E We A)
3	1, 2	sylan 343	1 |- ((Ord A /\ (B (_ A /\ -. B = (/))) -> E.x e. B (B i^i x) = (/))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = wceq 1091  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707  Ecep 2056   We wwe 2062  Ord word 2198

This theorem is referenced by:  tz7.7 2224  tfi 2244  onint 2261  peano5 2394

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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