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Theorem htalem 8703
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 473 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝑅 We 𝐴)
3 htalem.1 . . . . 5 𝐴 ∈ V
43a1i 11 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ∈ V)
5 ssid 3603 . . . . 5 𝐴𝐴
65a1i 11 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
7 simpr 477 . . . 4 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐴 ≠ ∅)
8 wereu 5070 . . . 4 ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴𝐴𝐴 ≠ ∅)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
92, 4, 6, 7, 8syl13anc 1325 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
10 riotacl 6579 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
119, 10syl 17 . 2 ((𝑅 We 𝐴𝐴 ≠ ∅) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
121, 11syl5eqel 2702 1 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  ∃!wreu 2909  Vcvv 3186  wss 3555  c0 3891   class class class wbr 4613   We wwe 5032  crio 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-iota 5810  df-riota 6565
This theorem is referenced by:  hta  8704
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