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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 2819 | . . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | eqid 2819 | . . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵 | |
3 | 1, 2 | fnebas 33685 | . . 3 ⊢ (𝐴Fne𝐵 → ∪ 𝐴 = ∪ 𝐵) |
4 | eqid 2819 | . . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
5 | 2, 4 | fnebas 33685 | . . 3 ⊢ (𝐵Fne𝐶 → ∪ 𝐵 = ∪ 𝐶) |
6 | 3, 5 | sylan9eq 2874 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → ∪ 𝐴 = ∪ 𝐶) |
7 | fnerel 33679 | . . . . 5 ⊢ Rel Fne | |
8 | 7 | brrelex2i 5602 | . . . 4 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
9 | 1, 2 | isfne4b 33682 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
10 | 9 | simplbda 502 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴Fne𝐵) → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
11 | 8, 10 | mpancom 686 | . . 3 ⊢ (𝐴Fne𝐵 → (topGen‘𝐴) ⊆ (topGen‘𝐵)) |
12 | 7 | brrelex2i 5602 | . . . 4 ⊢ (𝐵Fne𝐶 → 𝐶 ∈ V) |
13 | 2, 4 | isfne4b 33682 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐵Fne𝐶 ↔ (∪ 𝐵 = ∪ 𝐶 ∧ (topGen‘𝐵) ⊆ (topGen‘𝐶)))) |
14 | 13 | simplbda 502 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵Fne𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
15 | 12, 14 | mpancom 686 | . . 3 ⊢ (𝐵Fne𝐶 → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
16 | 11, 15 | sylan9ss 3978 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → (topGen‘𝐴) ⊆ (topGen‘𝐶)) |
17 | 12 | adantl 484 | . . 3 ⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐶 ∈ V) |
18 | 1, 4 | isfne4b 33682 | . . 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 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ⊆ wss 3934 ∪ cuni 4830 class class class wbr 5057 ‘cfv 6348 topGenctg 16703 Fnecfne 33677 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16709 df-fne 33678 |
This theorem is referenced by: fnessref 33698 fnemeet2 33708 fnejoin2 33710 |
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