Theorem htalem 9891

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 484	. . . 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 4006	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐴)
6		simpr 486	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
7		wereu 5673	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
8	2, 4, 5, 6, 7	syl13anc 1373	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
9		riotacl 7383	. . 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 2838	1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃!wreu 3375  Vcvv 3475   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149   We wwe 5631  ℩crio 7364

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-iota 6496  df-riota 7365

This theorem is referenced by:  hta  9892