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Theorem fness 35734
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 1135 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝑋 = 𝑌)
2 ssel2 3970 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
323adant3 1129 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑥𝐵)
4 simp3 1135 . . . . . . 7 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑦𝑥)
5 ssid 3997 . . . . . . 7 𝑥𝑥
64, 5jctir 520 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → (𝑦𝑥𝑥𝑥))
7 elequ2 2113 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
8 sseq1 4000 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
97, 8anbi12d 630 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
109rspcev 3604 . . . . . 6 ((𝑥𝐵 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
113, 6, 10syl2anc 583 . . . . 5 ((𝐴𝐵𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
12113expib 1119 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥)))
1312ralrimivv 3190 . . 3 (𝐴𝐵 → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
14133ad2ant2 1131 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
15 fness.1 . . . 4 𝑋 = 𝐴
16 fness.2 . . . 4 𝑌 = 𝐵
1715, 16isfne2 35727 . . 3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
18173ad2ant1 1130 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
191, 14, 18mpbir2and 710 1 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wrex 3062  wss 3941   cuni 4900   class class class wbr 5139  Fnecfne 35721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-topgen 17394  df-fne 35722
This theorem is referenced by:  fnessref  35742  refssfne  35743
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