Theorem domeng 3285

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 group(s):   x,A   x,B

Proof of Theorem domeng

Step	Hyp	Ref	Expression
1		breq2 2066	. . 3 |- (y = B -> (A ~<_ y <-> A ~<_ B))
2		sseq2 1522	. . . . 5 |- (y = B -> (x (_ y <-> x (_ B))
3	2	anbi2d 468	. . . 4 |- (y = B -> ((A ~~ x /\ x (_ y) <-> (A ~~ x /\ x (_ B)))
4	3	biexdv 936	. . 3 |- (y = B -> (E.x(A ~~ x /\ x (_ y) <-> E.x(A ~~ x /\ x (_ B)))
5	1, 4	bibi12d 477	. 2 |- (y = B -> ((A ~<_ y <-> E.x(A ~~ x /\ x (_ y)) <-> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))))
6		visset 1350	. . 3 |- y e. V
7	6	domen 3284	. 2 |- (A ~<_ y <-> E.x(A ~~ x /\ x (_ y))
8	5, 7	vtoclg 1383	1 |- (B e. C -> (A ~<_ B <-> E.x(A ~~ x /\ x (_ B)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092   (_ wss 1487   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272

This theorem is referenced by:  isfinite2 3437

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  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-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274  df-dom 3275

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