Theorem htalem 8335

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 antecedent. The element 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
htalem.2

Assertion

Ref	Expression
htalem

Distinct variable groups:   ,,   ,,

Proof of Theorem htalem

Step	Hyp	Ref	Expression
1		htalem.2	. 2
2		simpl 457	. . . 4
3		htalem.1	. . . . 5
4	3	a1i 11	. . . 4
5		ssid 3522	. . . . 5
6	5	a1i 11	. . . 4
7		simpr 461	. . . 4
8		wereu 4880	. . . 4
9	2, 4, 6, 7, 8	syl13anc 1230	. . 3
10		riotacl 6272	. . 3
11	9, 10	syl 16	. 2
12	1, 11	syl5eqel 2549	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E!wreu 2809  cvv 3109  C_wss 3475  c0 3784   class class class wbr 4452  Wewwe 4842  iota_crio 6256

This theorem is referenced by:  hta  8336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-iota 5556  df-riota 6257