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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 antecedent. The element 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
htalem.2
Assertion
Ref Expression
htalem
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem htalem
StepHypRef Expression
1 htalem.2 . 2
2 simpl 444 . . . 4
3 htalem.1 . . . . 5
43a1i 11 . . . 4
5 ssid 3359 . . . . 5
65a1i 11 . . . 4
7 simpr 448 . . . 4
8 wereu 4570 . . . 4
92, 4, 6, 7, 8syl13anc 1186 . . 3
10 riotacl 6556 . . 3
119, 10syl 16 . 2
121, 11syl5eqel 2519 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  wreu 2699  cvv 2948   wss 3312  c0 3620   class class class wbr 4204   wwe 4532  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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