Theorem fnessex 36644

Description: If 𝐵 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐵. (Contributed by Jeff Hankins, 28-Sep-2009.)

Assertion

Ref	Expression
fnessex	⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑃   𝑥,𝑆

Proof of Theorem fnessex

Step	Hyp	Ref	Expression
1		fnetg 36643	. . 3 ⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵))
2	1	sselda 3927	. 2 ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → 𝑆 ∈ (topGen‘𝐵))
3		tg2 22994	. 2 ⊢ ((𝑆 ∈ (topGen‘𝐵) ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆))
4	2, 3	stoic3 1786	1 ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   ∈ wcel 2132  ∃wrex 3076   ⊆ wss 3895   class class class wbr 5090  ‘cfv 6506  topGenctg 17438  Fnecfne 36634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-iota 6462  df-fun 6508  df-fv 6514  df-topgen 17444  df-fne 36635

This theorem is referenced by:  fneint  36646  fnessref  36655