![]() |
Mathbox for Jeff Hankins |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fness | Structured version Visualization version GIF version |
Description: A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.) |
Ref | Expression |
---|---|
fness.1 | ⊢ 𝑋 = ∪ 𝐴 |
fness.2 | ⊢ 𝑌 = ∪ 𝐵 |
Ref | Expression |
---|---|
fness | ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1136 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
2 | ssel2 3975 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
3 | 2 | 3adant3 1130 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐵) |
4 | simp3 1136 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) | |
5 | ssid 4002 | . . . . . . 7 ⊢ 𝑥 ⊆ 𝑥 | |
6 | 4, 5 | jctir 520 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) |
7 | elequ2 2114 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥)) | |
8 | sseq1 4005 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥)) | |
9 | 7, 8 | anbi12d 631 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥))) |
10 | 9 | rspcev 3609 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) |
11 | 3, 6, 10 | syl2anc 583 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) |
12 | 11 | 3expib 1120 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) |
13 | 12 | ralrimivv 3195 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) |
14 | 13 | 3ad2ant2 1132 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) |
15 | fness.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
16 | fness.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
17 | 15, 16 | isfne2 35826 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) |
18 | 17 | 3ad2ant1 1131 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) |
19 | 1, 14, 18 | mpbir2and 712 | 1 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 ⊆ wss 3947 ∪ cuni 4908 class class class wbr 5148 Fnecfne 35820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-topgen 17424 df-fne 35821 |
This theorem is referenced by: fnessref 35841 refssfne 35842 |
Copyright terms: Public domain | W3C validator |