Theorem htalem 8759

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 e. _V
htalem.2	|- B = ( iota_ x e. A A. y e. A -. y R x )

Assertion

Ref	Expression
htalem	|- ( ( R We A /\ A =/= (/) ) -> B e. 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 = ( iota_ x e. A A. y e. A -. y R x )
2		simpl 473	. . . 4 |- ( ( R We A /\ A =/= (/) ) -> R We A )
3		htalem.1	. . . . 5 |- A e. _V
4	3	a1i 11	. . . 4 |- ( ( R We A /\ A =/= (/) ) -> A e. _V )
5		ssid 3624	. . . . 5 |- A C_ A
6	5	a1i 11	. . . 4 |- ( ( R We A /\ A =/= (/) ) -> A C_ A )
7		simpr 477	. . . 4 |- ( ( R We A /\ A =/= (/) ) -> A =/= (/) )
8		wereu 5110	. . . 4 |- ( ( R We A /\ ( A e. _V /\ A C_ A /\ A =/= (/) ) ) -> E! x e. A A. y e. A -. y R x )
9	2, 4, 6, 7, 8	syl13anc 1328	. . 3 |- ( ( R We A /\ A =/= (/) ) -> E! x e. A A. y e. A -. y R x )
10		riotacl 6625	. . 3 |- ( E! x e. A A. y e. A -. y R x -> ( iota_ x e. A A. y e. A -. y R x ) e. A )
11	9, 10	syl 17	. 2 |- ( ( R We A /\ A =/= (/) ) -> ( iota_ x e. A A. y e. A -. y R x ) e. A )
12	1, 11	syl5eqel 2705	1 |- ( ( R We A /\ A =/= (/) ) -> B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E!wreu 2914   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   We wwe 5072   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-iota 5851  df-riota 6611

This theorem is referenced by:  hta  8760