Theorem tz7.5 2969

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 2943	. 2 |- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))
2		ordwe 2961	. 2 |- (Ord A -> E We A)
3	1, 2	syl3an1 859	1 |- ((Ord A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   =/= wne 1585  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280  Ecep 2830   We wwe 2916  Ord word 2947

This theorem is referenced by:  tz7.7 2973  onint 3006  tfi 3126  peano5 3153

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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