Theorem domeng 4362

Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146.

Assertion

Ref	Expression
domeng	|- (B e. C -> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B)))

Distinct variable groups:   x,A   x,B

Proof of Theorem domeng

Step	Hyp	Ref	Expression
1		breq2 2613	. 2 |- (y = B -> (A ~<_ y <-> A ~<_ B))
2		sseq2 2073	. . . 4 |- (y = B -> (x (_ y <-> x (_ B))
3	2	anbi2d 614	. . 3 |- (y = B -> ((A ~~ x /\ x (_ y) <-> (A ~~ x /\ x (_ B)))
4	3	exbidv 1274	. 2 |- (y = B -> (E.x(A ~~ x /\ x (_ y) <-> E.x(A ~~ x /\ x (_ B)))
5		visset 1804	. . 3 |- y e. V
6	5	domen 4361	. 2 |- (A ~<_ y <-> E.x(A ~~ x /\ x (_ y))
7	1, 4, 6	vtoclbg 1839	1 |- (B e. C -> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   (_ wss 2037   class class class wbr 2609   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  domfi 4516  isfinite2 4523

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351  df-dom 4352

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