Theorem domeng 7036

Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem domeng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4134	. 2 |- ( y = B -> ( A ~<_ y <-> A ~<_ B ) )
2		sseq2 3272	. . . 4 |- ( y = B -> ( x C_ y <-> x C_ B ) )
3	2	anbi2d 468	. . 3 |- ( y = B -> ( ( A ~~ x /\ x C_ y ) <-> ( A ~~ x /\ x C_ B ) ) )
4	3	exbidv 1878	. 2 |- ( y = B -> ( E. x ( A ~~ x /\ x C_ y ) <-> E. x ( A ~~ x /\ x C_ B ) ) )
5		vex 2824	. . 3 |- y e. _V
6	5	domen 7035	. 2 |- ( A ~<_ y <-> E. x ( A ~~ x /\ x C_ y ) )
7	1, 4, 6	vtoclbg 2884	1 |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   class class class wbr 4130   ~~ cen 7020   ~<_ cdom 7021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-en 7023  df-dom 7024

This theorem is used by:  mapdom1g  7147