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Theorem fness 32571
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 1130 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝑋 = 𝑌)
2 ssel2 3704 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
323adant3 1124 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑥𝐵)
4 simp3 1130 . . . . . . 7 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑦𝑥)
5 ssid 3730 . . . . . . 7 𝑥𝑥
64, 5jctir 562 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → (𝑦𝑥𝑥𝑥))
7 elequ2 2117 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
8 sseq1 3732 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
97, 8anbi12d 749 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
109rspcev 3413 . . . . . 6 ((𝑥𝐵 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
113, 6, 10syl2anc 696 . . . . 5 ((𝐴𝐵𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
12113expib 1116 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥)))
1312ralrimivv 3072 . . 3 (𝐴𝐵 → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
14133ad2ant2 1126 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
15 fness.1 . . . 4 𝑋 = 𝐴
16 fness.2 . . . 4 𝑌 = 𝐵
1715, 16isfne2 32564 . . 3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
18173ad2ant1 1125 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
191, 14, 18mpbir2and 995 1 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  wrex 3015  wss 3680   cuni 4544   class class class wbr 4760  Fnecfne 32558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-topgen 16227  df-fne 32559
This theorem is referenced by:  fnessref  32579  refssfne  32580
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