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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnetr | Structured version Visualization version Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 | |
2 | eqid 2622 | . . . 4 | |
3 | 1, 2 | fnebas 32339 | . . 3 |
4 | eqid 2622 | . . . 4 | |
5 | 2, 4 | fnebas 32339 | . . 3 |
6 | 3, 5 | sylan9eq 2676 | . 2 |
7 | fnerel 32333 | . . . . 5 | |
8 | 7 | brrelex2i 5159 | . . . 4 |
9 | 1, 2 | isfne4b 32336 | . . . . 5 |
10 | 9 | simplbda 654 | . . . 4 |
11 | 8, 10 | mpancom 703 | . . 3 |
12 | 7 | brrelex2i 5159 | . . . 4 |
13 | 2, 4 | isfne4b 32336 | . . . . 5 |
14 | 13 | simplbda 654 | . . . 4 |
15 | 12, 14 | mpancom 703 | . . 3 |
16 | 11, 15 | sylan9ss 3616 | . 2 |
17 | 12 | adantl 482 | . . 3 |
18 | 1, 4 | isfne4b 32336 | . . 3 |
19 | 17, 18 | syl 17 | . 2 |
20 | 6, 16, 19 | mpbir2and 957 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 cuni 4436 class class class wbr 4653 cfv 5888 ctg 16098 cfne 32331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 df-fne 32332 |
This theorem is referenced by: fnessref 32352 fnemeet2 32362 fnejoin2 32364 |
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