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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 2725 | . . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | eqid 2725 | . . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵 | |
3 | 1, 2 | fnebas 35959 | . . 3 ⊢ (𝐴Fne𝐵 → ∪ 𝐴 = ∪ 𝐵) |
4 | eqid 2725 | . . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
5 | 2, 4 | fnebas 35959 | . . 3 ⊢ (𝐵Fne𝐶 → ∪ 𝐵 = ∪ 𝐶) |
6 | 3, 5 | sylan9eq 2785 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → ∪ 𝐴 = ∪ 𝐶) |
7 | fnerel 35953 | . . . . 5 ⊢ Rel Fne | |
8 | 7 | brrelex2i 5735 | . . . 4 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
9 | 1, 2 | isfne4b 35956 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
10 | 9 | simplbda 498 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴Fne𝐵) → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
11 | 8, 10 | mpancom 686 | . . 3 ⊢ (𝐴Fne𝐵 → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
12 | 7 | brrelex2i 5735 | . . . 4 ⊢ (𝐵Fne𝐶 → 𝐶 ∈ V) |
13 | 2, 4 | isfne4b 35956 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐵Fne𝐶 ↔ (∪ 𝐵 = ∪ 𝐶 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐶)))) |
14 | 13 | simplbda 498 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵Fne𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
15 | 12, 14 | mpancom 686 | . . 3 ⊢ (𝐵Fne𝐶 → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
16 | 11, 15 | sylan9ss 3990 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → (topGen‘𝐴) ⊆ (topGen‘𝐶)) |
17 | 12 | adantl 480 | . . 3 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐶 ∈ V) |
18 | 1, 4 | isfne4b 35956 | . . 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 711 | 1 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 ‘cfv 6549 topGenctg 17422 Fnecfne 35951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-topgen 17428 df-fne 35952 |
This theorem is referenced by: fnessref 35972 fnemeet2 35982 fnejoin2 35984 |
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