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Theorem fness 33810
 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 3910 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
323adant3 1129 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑥𝐵)
4 simp3 1135 . . . . . . 7 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑦𝑥)
5 ssid 3937 . . . . . . 7 𝑥𝑥
64, 5jctir 524 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → (𝑦𝑥𝑥𝑥))
7 elequ2 2126 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
8 sseq1 3940 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
97, 8anbi12d 633 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
109rspcev 3571 . . . . . 6 ((𝑥𝐵 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
113, 6, 10syl2anc 587 . . . . 5 ((𝐴𝐵𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
12113expib 1119 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥)))
1312ralrimivv 3155 . . 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 33803 . . 3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
18173ad2ant1 1130 . 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ cuni 4800   class class class wbr 5030  Fnecfne 33797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-topgen 16709  df-fne 33798 This theorem is referenced by:  fnessref  33818  refssfne  33819
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