MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  htalem Structured version   Visualization version   GIF version

Theorem htalem 9934
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 ∈ V
htalem.2 𝐵 = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
Assertion
Ref Expression
htalem ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
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 = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
2 simpl 482 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝑅 We 𝐴)
3 htalem.1 . . . . 5 𝐴 ∈ V
43a1i 11 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ∈ V)
5 ssidd 4019 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
6 simpr 484 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7 wereu 5685 . . . 4 ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴𝐴𝐴 ≠ ∅)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
82, 4, 5, 6, 7syl13anc 1371 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
9 riotacl 7405 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
108, 9syl 17 . 2 ((𝑅 We 𝐴𝐴 ≠ ∅) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
111, 10eqeltrid 2843 1 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  ∃!wreu 3376  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148   We wwe 5640  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-iota 6516  df-riota 7388
This theorem is referenced by:  hta  9935
  Copyright terms: Public domain W3C validator