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Theorem htalem 7755
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 3312 . . . . 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 4521 . . . 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 6502 . . 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 2473 1  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E!wreu 2653   _Vcvv 2901    C_ wss 3265   (/)c0 3573   class class class wbr 4155    We wwe 4483   iota_crio 6480
This theorem is referenced by:  hta  7756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-iota 5360  df-riota 6487
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