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Theorem htalem 9893
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 481 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝑅 We 𝐴)
3 htalem.1 . . . . 5 𝐴 ∈ V
43a1i 11 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ∈ V)
5 ssidd 4004 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
6 simpr 483 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7 wereu 5671 . . . 4 ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴𝐴𝐴 ≠ ∅)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
82, 4, 5, 6, 7syl13anc 1370 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
9 riotacl 7385 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
108, 9syl 17 . 2 ((𝑅 We 𝐴𝐴 ≠ ∅) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
111, 10eqeltrid 2835 1 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  wne 2938  wral 3059  ∃!wreu 3372  Vcvv 3472  wss 3947  c0 4321   class class class wbr 5147   We wwe 5629  crio 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-iota 6494  df-riota 7367
This theorem is referenced by:  hta  9894
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