Theorem htalem 9512

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 486	. . . 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 3924	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐴)
6		simpr 488	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7		wereu 5547	. . . 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 7188	. . 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 2842	1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃!wreu 3063  Vcvv 3408   ⊆ wss 3866  ∅c0 4237   class class class wbr 5053   We wwe 5508  ℩crio 7169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-iota 6338  df-riota 7170

This theorem is referenced by:  hta  9513