Theorem htalem 9881

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 487	. . . 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 3968	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐴)
6		simpr 489	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7		wereu 5658	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
8	2, 4, 5, 6, 7	syl13anc 1397	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
9		riotacl 7385	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
10	8, 9	syl 18	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ∈ 𝐴)
11	1, 10	eqeltrid 2873	1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃!wreu 3374  Vcvv 3463   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113   We wwe 5614  ℩crio 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-iota 6493  df-riota 7368

This theorem is referenced by:  hta  9882