Theorem htalem 9654

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 483	. . . 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 3944	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐴)
6		simpr 485	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7		wereu 5585	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
8	2, 4, 5, 6, 7	syl13anc 1371	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
9		riotacl 7250	. . 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 2843	1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃!wreu 3066  Vcvv 3432   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074   We wwe 5543  ℩crio 7231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-iota 6391  df-riota 7232

This theorem is referenced by:  hta  9655