Theorem dmhmpha 10515

Description: A simple consequence of dmhmph 10513. A sort of lemma for hmpher 10517 since it is used several times by it.

Hypothesis

Ref	Expression
dmhmpha.1	|- A e. V

Assertion

Ref	Expression
dmhmpha	|- (A ~= B -> A e. Top)

Proof of Theorem dmhmpha

Step	Hyp	Ref	Expression
1		dmhmpha.1	. . 3 |- A e. V
2	1	breldm 3312	. 2 |- (A ~= B -> A e. dom ~= )
3		dmhmph 10513	. . 3 |- dom ~= (_ Top
4	3	sseli 2063	. 2 |- (A e. dom ~= -> A e. Top)
5	2, 4	syl 10	1 |- (A ~= B -> A e. Top)

Syntax hints:   -> wi 3   e. wcel 957  Vcvv 1809   class class class wbr 2616  dom cdm 3167  Topctop 7567   ~= chomeo 10495

This theorem is referenced by:  hmpher 10517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-xp 3181  df-dm 3185  df-hmph 10504

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