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Theorem htalem 9327
 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 486 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝑅 We 𝐴)
3 htalem.1 . . . . 5 𝐴 ∈ V
43a1i 11 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ∈ V)
5 ssidd 3940 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
6 simpr 488 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7 wereu 5519 . . . 4 ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴𝐴𝐴 ≠ ∅)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
82, 4, 5, 6, 7syl13anc 1369 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
9 riotacl 7120 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
108, 9syl 17 . 2 ((𝑅 We 𝐴𝐴 ≠ ∅) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
111, 10eqeltrid 2894 1 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃!wreu 3108  Vcvv 3442   ⊆ wss 3883  ∅c0 4246   class class class wbr 5034   We wwe 5481  ℩crio 7102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-iota 6291  df-riota 7103 This theorem is referenced by:  hta  9328
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