Theorem oe0lem 4158

Description: A helper lemma for oe0 4167 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 373	. . 3 |- (ph -> (A = (/) -> ps))
3	2	adantl 390	. 2 |- ((A e. On /\ ph) -> (A = (/) -> ps))
4		on0eln0 3030	. . . 4 |- (A e. On -> ((/) e. A <-> A =/= (/)))
5	4	adantr 391	. . 3 |- ((A e. On /\ ph) -> ((/) e. A <-> A =/= (/)))
6		oe0lem.2	. . . 4 |- (((A e. On /\ ph) /\ (/) e. A) -> ps)
7	6	ex 373	. . 3 |- ((A e. On /\ ph) -> ((/) e. A -> ps))
8	5, 7	sylbird 205	. 2 |- ((A e. On /\ ph) -> (A =/= (/) -> ps))
9	3, 8	pm2.61dne 1638	1 |- ((A e. On /\ ph) -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  (/)c0 2283  Oncon0 2954

This theorem is referenced by:  oe0 4167  oev2 4168  oesuc 4172  oecl 4178  odi 4216  oewordri 4225  oelim2 4228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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