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Theorem htalem 7812
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 444 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  R  We  A )
3 htalem.1 . . . . 5  |-  A  e. 
_V
43a1i 11 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  e.  _V )
5 ssid 3359 . . . . 5  |-  A  C_  A
65a1i 11 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  C_  A )
7 simpr 448 . . . 4  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  A  =/=  (/) )
8 wereu 4570 . . . 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 1186 . . 3  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  E! x  e.  A  A. y  e.  A  -.  y R x )
10 riotacl 6556 . . 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 16 . 2  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  ( iota_ x  e.  A A. y  e.  A  -.  y R x )  e.  A )
121, 11syl5eqel 2519 1  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E!wreu 2699   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204    We wwe 4532   iota_crio 6534
This theorem is referenced by:  hta  7813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-iota 5410  df-riota 6541
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