Theorem oe0lem 5197

Description: A helper lemma for oe0 5206 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	ex 402	. . 3 |- (ph -> (A = (/) -> ps))
3	2	adantl 424	. 2 |- ((A e. On /\ ph) -> (A = (/) -> ps))
4		on0eln0 3718	. . . 4 |- (A e. On -> ((/) e. A <-> A =/= (/)))
5	4	adantr 425	. . 3 |- ((A e. On /\ ph) -> ((/) e. A <-> A =/= (/)))
6		oe0lem.2	. . . 4 |- (((A e. On /\ ph) /\ (/) e. A) -> ps)
7	6	ex 402	. . 3 |- ((A e. On /\ ph) -> ((/) e. A -> ps))
8	5, 7	sylbird 222	. 2 |- ((A e. On /\ ph) -> (A =/= (/) -> ps))
9	3, 8	pm2.61dne 2091	1 |- ((A e. On /\ ph) -> ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875  Oncon0 3657

This theorem is referenced by:  oe0 5206  oev2 5207  oesuc 5211  oecl 5218  oeclOLD 5219  odi 5258  oewordri 5267  oelim2 5270  oeoa 5272  oeoe 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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