Theorem htalem 9849

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 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)

Assertion

Ref	Expression
htalem	⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem htalem

Step	Hyp	Ref	Expression
1		htalem.2	. 2 ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
2		simpl 482	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝑅 We 𝐴)
3		htalem.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	a1i 11	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
5		ssidd 3970	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐴)
6		simpr 484	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7		wereu 5634	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
8	2, 4, 5, 6, 7	syl13anc 1374	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
9		riotacl 7361	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
10	8, 9	syl 17	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
11	1, 10	eqeltrid 2832	1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃!wreu 3352  Vcvv 3447   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107   We wwe 5590  ℩crio 7343

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-iota 6464  df-riota 7344

This theorem is referenced by:  hta  9850