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Theorem htalem 7566
Description: Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional  R  We  A antecedent. The element  B is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
htalem.1  |-  A  e. 
_V
htalem.2  |-  B  =  ( iota_ x  e.  A A. y  e.  A  -.  y R x )
Assertion
Ref Expression
htalem  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
Distinct variable groups:    x, y, A    x, R, y
Allowed substitution hints:    B( x, y)

Proof of Theorem htalem
StepHypRef Expression
1 htalem.2 . 2  |-  B  =  ( iota_ x  e.  A A. y  e.  A  -.  y R x )
2 simpl 443 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  R  We  A )
3 htalem.1 . . . . 5  |-  A  e. 
_V
43a1i 10 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  e.  _V )
5 ssid 3197 . . . . 5  |-  A  C_  A
65a1i 10 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  C_  A )
7 simpr 447 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  =/=  (/) )
8 wereu 4389 . . . 4  |-  ( ( R  We  A  /\  ( A  e.  _V  /\  A  C_  A  /\  A  =/=  (/) ) )  ->  E! x  e.  A  A. y  e.  A  -.  y R x )
92, 4, 6, 7, 8syl13anc 1184 . . 3  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  E! x  e.  A  A. y  e.  A  -.  y R x )
10 riotacl 6319 . . 3  |-  ( E! x  e.  A  A. y  e.  A  -.  y R x  ->  ( iota_ x  e.  A A. y  e.  A  -.  y R x )  e.  A )
119, 10syl 15 . 2  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  ( iota_ x  e.  A A. y  e.  A  -.  y R x )  e.  A )
121, 11syl5eqel 2367 1  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E!wreu 2545   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023    We wwe 4351   iota_crio 6297
This theorem is referenced by:  hta  7567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-iota 5219  df-riota 6304
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