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Theorem fness 31983
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 1061 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝑋 = 𝑌)
2 ssel2 3578 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
323adant3 1079 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑥𝐵)
4 simp3 1061 . . . . . . 7 ((𝐴𝐵𝑥𝐴𝑦𝑥) → 𝑦𝑥)
5 ssid 3603 . . . . . . 7 𝑥𝑥
64, 5jctir 560 . . . . . 6 ((𝐴𝐵𝑥𝐴𝑦𝑥) → (𝑦𝑥𝑥𝑥))
7 elequ2 2001 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
8 sseq1 3605 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑥𝑥𝑥))
97, 8anbi12d 746 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦𝑧𝑧𝑥) ↔ (𝑦𝑥𝑥𝑥)))
109rspcev 3295 . . . . . 6 ((𝑥𝐵 ∧ (𝑦𝑥𝑥𝑥)) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
113, 6, 10syl2anc 692 . . . . 5 ((𝐴𝐵𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
12113expib 1265 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥) → ∃𝑧𝐵 (𝑦𝑧𝑧𝑥)))
1312ralrimivv 2964 . . 3 (𝐴𝐵 → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
14133ad2ant2 1081 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))
15 fness.1 . . . 4 𝑋 = 𝐴
16 fness.2 . . . 4 𝑌 = 𝐵
1715, 16isfne2 31976 . . 3 (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
18173ad2ant1 1080 . 2 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
191, 14, 18mpbir2and 956 1 ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555   cuni 4402   class class class wbr 4613  Fnecfne 31970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025  df-fne 31971
This theorem is referenced by:  fnessref  31991  refssfne  31992
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