![]() |
Mathbox for Jeff Hankins |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fnetr | Structured version Visualization version GIF version |
Description: Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fnetr | ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | eqid 2735 | . . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵 | |
3 | 1, 2 | fnebas 36327 | . . 3 ⊢ (𝐴Fne𝐵 → ∪ 𝐴 = ∪ 𝐵) |
4 | eqid 2735 | . . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
5 | 2, 4 | fnebas 36327 | . . 3 ⊢ (𝐵Fne𝐶 → ∪ 𝐵 = ∪ 𝐶) |
6 | 3, 5 | sylan9eq 2795 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → ∪ 𝐴 = ∪ 𝐶) |
7 | fnerel 36321 | . . . . 5 ⊢ Rel Fne | |
8 | 7 | brrelex2i 5746 | . . . 4 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
9 | 1, 2 | isfne4b 36324 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
10 | 9 | simplbda 499 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴Fne𝐵) → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
11 | 8, 10 | mpancom 688 | . . 3 ⊢ (𝐴Fne𝐵 → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
12 | 7 | brrelex2i 5746 | . . . 4 ⊢ (𝐵Fne𝐶 → 𝐶 ∈ V) |
13 | 2, 4 | isfne4b 36324 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐵Fne𝐶 ↔ (∪ 𝐵 = ∪ 𝐶 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐶)))) |
14 | 13 | simplbda 499 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵Fne𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
15 | 12, 14 | mpancom 688 | . . 3 ⊢ (𝐵Fne𝐶 → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
16 | 11, 15 | sylan9ss 4009 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → (topGen‘𝐴) ⊆ (topGen‘𝐶)) |
17 | 12 | adantl 481 | . . 3 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐶 ∈ V) |
18 | 1, 4 | isfne4b 36324 | . . 3 ⊢ (𝐶 ∈ V → (𝐴Fne𝐶 ↔ (∪ 𝐴 = ∪ 𝐶 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐶)))) |
19 | 17, 18 | syl 17 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → (𝐴Fne𝐶 ↔ (∪ 𝐴 = ∪ 𝐶 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐶)))) |
20 | 6, 16, 19 | mpbir2and 713 | 1 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 ‘cfv 6563 topGenctg 17484 Fnecfne 36319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-topgen 17490 df-fne 36320 |
This theorem is referenced by: fnessref 36340 fnemeet2 36350 fnejoin2 36352 |
Copyright terms: Public domain | W3C validator |