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Theorem axcontlem3 25910
 Description: Lemma for axcont 25920. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)
Hypothesis
Ref Expression
axcontlem3.1
Assertion
Ref Expression
axcontlem3
Distinct variable groups:   ,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   ()   (,,)

Proof of Theorem axcontlem3
StepHypRef Expression
1 simplr2 1001 . 2
2 simpr2 965 . . . . . 6
32anim1i 553 . . . . 5
4 simplr3 1002 . . . . . 6
54adantr 453 . . . . 5
6 breq1 4218 . . . . . 6
7 opeq2 3987 . . . . . . 7
87breq2d 4227 . . . . . 6
96, 8rspc2v 3060 . . . . 5
103, 5, 9sylc 59 . . . 4
1110orcd 383 . . 3
1211ralrimiva 2791 . 2
13 axcontlem3.1 . . . 4
1413sseq2i 3375 . . 3
15 ssrab 3423 . . 3
1614, 15bitri 242 . 2
171, 12, 16sylanbrc 647 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 359   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601  wral 2707  crab 2711   wss 3322  cop 3819   class class class wbr 4215  cfv 5457  cn 10005  cee 25832   cbtwn 25833 This theorem is referenced by:  axcontlem9  25916  axcontlem10  25917 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216
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