Theorem oe0lem 3121

Description: A helper lemma for oe0 3130 and others.

Hypotheses

Ref	Expression
oe0lem.1	|- ((ph /\ A = (/)) -> ps)
oe0lem.2	|- (((A e. On /\ ph) /\ (/) e. A) -> ps)

Assertion

Ref	Expression
oe0lem	|- ((A e. On /\ ph) -> ps)

Proof of Theorem oe0lem

Step	Hyp	Ref	Expression
1		oe0lem.1	. . . 4 |- ((ph /\ A = (/)) -> ps)
2	1	exp 291	. . 3 |- (ph -> (A = (/) -> ps))
3	2	adantl 305	. 2 |- ((A e. On /\ ph) -> (A = (/) -> ps))
4		on0eln0 2279	. . . 4 |- (A e. On -> ((/) e. A <-> -. A = (/)))
5	4	adantr 306	. . 3 |- ((A e. On /\ ph) -> ((/) e. A <-> -. A = (/)))
6		oe0lem.2	. . . 4 |- (((A e. On /\ ph) /\ (/) e. A) -> ps)
7	6	exp 291	. . 3 |- ((A e. On /\ ph) -> ((/) e. A -> ps))
8	5, 7	sylbird 180	. 2 |- ((A e. On /\ ph) -> (-. A = (/) -> ps))
9	3, 8	pm2.61d 112	1 |- ((A e. On /\ ph) -> ps)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  (/)c0 1707  Oncon0 2199

This theorem is referenced by:  oe0 3130  oesuc 3134  oecl 3140

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  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-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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