Theorem htalem 3618

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.

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 group(s):   x,y,A   x,R,y

Proof of Theorem htalem

Step	Hyp	Ref	Expression
1		ssid 1519	. . . 4 ⊢ A ⊆ A
2		htalem.1	. . . . 5 ⊢ A ∈ V
3	2	wereu 2197	. . . 4 ⊢ ((R We A ∧ (A ⊆ A ∧ ¬ A = ∅)) → ∃!x ∈ A ∀y ∈ A ¬ yRx)
4	1, 3	mpan21 531	. . 3 ⊢ ((R We A ∧ ¬ A = ∅) → ∃!x ∈ A ∀y ∈ A ¬ yRx)
5		reucl 1957	. . 3 ⊢ (∃!x ∈ A ∀y ∈ A ¬ yRx → ∪{x ∈ A∣∀y ∈ A ¬ yRx} ∈ A)
6	4, 5	syl 12	. 2 ⊢ ((R We A ∧ ¬ A = ∅) → ∪{x ∈ A∣∀y ∈ A ¬ yRx} ∈ A)
7		htalem.2	. . 3 ⊢ B = ∪{x ∈ A∣∀y ∈ A ¬ yRx}
8	7	eleq1i 1152	. 2 ⊢ (B ∈ A ↔ ∪{x ∈ A∣∀y ∈ A ¬ yRx} ∈ A)
9	6, 8	sylibr 175	1 ⊢ ((R We A ∧ ¬ A = ∅) → B ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃!wreu 1203  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919   class class class wbr 2054   We wwe 2062

This theorem is referenced by:  hta 3619

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186

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