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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 2610 | . . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | eqid 2610 | . . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵 | |
3 | 1, 2 | fnebas 31509 | . . 3 ⊢ (𝐴Fne𝐵 → ∪ 𝐴 = ∪ 𝐵) |
4 | eqid 2610 | . . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
5 | 2, 4 | fnebas 31509 | . . 3 ⊢ (𝐵Fne𝐶 → ∪ 𝐵 = ∪ 𝐶) |
6 | 3, 5 | sylan9eq 2664 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → ∪ 𝐴 = ∪ 𝐶) |
7 | fnerel 31503 | . . . . 5 ⊢ Rel Fne | |
8 | 7 | brrelex2i 5083 | . . . 4 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
9 | 1, 2 | isfne4b 31506 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
10 | 9 | simplbda 652 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴Fne𝐵) → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
11 | 8, 10 | mpancom 700 | . . 3 ⊢ (𝐴Fne𝐵 → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
12 | 7 | brrelex2i 5083 | . . . 4 ⊢ (𝐵Fne𝐶 → 𝐶 ∈ V) |
13 | 2, 4 | isfne4b 31506 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐵Fne𝐶 ↔ (∪ 𝐵 = ∪ 𝐶 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐶)))) |
14 | 13 | simplbda 652 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵Fne𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
15 | 12, 14 | mpancom 700 | . . 3 ⊢ (𝐵Fne𝐶 → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
16 | 11, 15 | sylan9ss 3581 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → (topGen‘𝐴) ⊆ (topGen‘𝐶)) |
17 | 12 | adantl 481 | . . 3 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐶 ∈ V) |
18 | 1, 4 | isfne4b 31506 | . . 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 959 | 1 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 topGenctg 15921 Fnecfne 31501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 df-fne 31502 |
This theorem is referenced by: fnessref 31522 fnemeet2 31532 fnejoin2 31534 |
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