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Theorem fness 35833
Description: A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
Hypotheses
Ref Expression
fness.1 𝑋 = 𝐴
fness.2 𝑌 = 𝐵
Assertion
Ref Expression
fness ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)

Proof of Theorem fness
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1136 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝑋 = 𝑌)
2 ssel2 3975 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
323adant3 1130 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑥𝐵)
4 simp3 1136 . . . . . . 7 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑦𝑥)
5 ssid 4002 . . . . . . 7 𝑥𝑥
64, 5jctir 520 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → (𝑦𝑥𝑥𝑥))
7 elequ2 2114 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
8 sseq1 4005 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
97, 8anbi12d 631 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
109rspcev 3609 . . . . . 6 ((𝑥𝐵 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
113, 6, 10syl2anc 583 . . . . 5 ((𝐴𝐵𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
12113expib 1120 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥)))
1312ralrimivv 3195 . . 3 (𝐴𝐵 → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
14133ad2ant2 1132 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
15 fness.1 . . . 4 𝑋 = 𝐴
16 fness.2 . . . 4 𝑌 = 𝐵
1715, 16isfne2 35826 . . 3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
18173ad2ant1 1131 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
191, 14, 18mpbir2and 712 1 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  wrex 3067  wss 3947   cuni 4908   class class class wbr 5148  Fnecfne 35820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topgen 17424  df-fne 35821
This theorem is referenced by:  fnessref  35841  refssfne  35842
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