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Mirrors > Home > NFE Home > Th. List > uniiunlem | Unicode version |
Description: A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
uniiunlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2359 | . . . . . 6 | |
2 | 1 | rexbidv 2635 | . . . . 5 |
3 | 2 | cbvabv 2472 | . . . 4 |
4 | 3 | sseq1i 3295 | . . 3 |
5 | r19.23v 2730 | . . . . 5 | |
6 | 5 | albii 1566 | . . . 4 |
7 | ralcom4 2877 | . . . 4 | |
8 | abss 3335 | . . . 4 | |
9 | 6, 7, 8 | 3bitr4i 268 | . . 3 |
10 | 4, 9 | bitr4i 243 | . 2 |
11 | nfv 1619 | . . . . 5 | |
12 | eleq1 2413 | . . . . 5 | |
13 | 11, 12 | ceqsalg 2883 | . . . 4 |
14 | 13 | ralimi 2689 | . . 3 |
15 | ralbi 2750 | . . 3 | |
16 | 14, 15 | syl 15 | . 2 |
17 | 10, 16 | syl5rbb 249 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wal 1540 wceq 1642 wcel 1710 cab 2339 wral 2614 wrex 2615 wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: (None) |
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