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Theorem isfne 26339
 Description: The predicate " is finer than ." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
isfne.1
isfne.2
Assertion
Ref Expression
isfne
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem isfne
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 26338 . . . . 5
21brrelexi 4910 . . . 4
32anim1i 552 . . 3
43ancoms 440 . 2
5 simpr 448 . . . . 5
6 isfne.1 . . . . 5
7 isfne.2 . . . . 5
85, 6, 73eqtr3g 2490 . . . 4
9 simpr 448 . . . . . . 7
10 uniexg 4698 . . . . . . . 8
1110adantr 452 . . . . . . 7
129, 11eqeltrd 2509 . . . . . 6
13 uniexb 4744 . . . . . 6
1412, 13sylibr 204 . . . . 5
15 simpl 444 . . . . 5
1614, 15jca 519 . . . 4
178, 16syldan 457 . . 3
19 unieq 4016 . . . . . 6
2019, 6syl6eqr 2485 . . . . 5
2120eqeq1d 2443 . . . 4
22 raleq 2896 . . . 4
2321, 22anbi12d 692 . . 3
24 unieq 4016 . . . . . 6
2524, 7syl6eqr 2485 . . . . 5
2625eqeq2d 2446 . . . 4
27 ineq1 3527 . . . . . . 7
2827unieqd 4018 . . . . . 6
2928sseq2d 3368 . . . . 5
3029ralbidv 2717 . . . 4
3126, 30anbi12d 692 . . 3
32 df-fne 26334 . . 3
3323, 31, 32brabg 4466 . 2
344, 18, 33pm5.21nd 869 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948   cin 3311   wss 3312  cpw 3791  cuni 4007   class class class wbr 4204  cfne 26330 This theorem is referenced by:  isfne4  26340 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-fne 26334
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