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Mirrors > Home > ILE Home > Th. List > exss | Unicode version |
Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.) |
Ref | Expression |
---|---|
exss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabn0m 3390 | . . 3 | |
2 | df-rab 2425 | . . . . 5 | |
3 | 2 | eleq2i 2206 | . . . 4 |
4 | 3 | exbii 1584 | . . 3 |
5 | 1, 4 | bitr3i 185 | . 2 |
6 | vex 2689 | . . . . . 6 | |
7 | 6 | snss 3649 | . . . . 5 |
8 | ssab2 3181 | . . . . . 6 | |
9 | sstr2 3104 | . . . . . 6 | |
10 | 8, 9 | mpi 15 | . . . . 5 |
11 | 7, 10 | sylbi 120 | . . . 4 |
12 | simpr 109 | . . . . . . . 8 | |
13 | equsb1 1758 | . . . . . . . . 9 | |
14 | velsn 3544 | . . . . . . . . . 10 | |
15 | 14 | sbbii 1738 | . . . . . . . . 9 |
16 | 13, 15 | mpbir 145 | . . . . . . . 8 |
17 | 12, 16 | jctil 310 | . . . . . . 7 |
18 | df-clab 2126 | . . . . . . . 8 | |
19 | sban 1928 | . . . . . . . 8 | |
20 | 18, 19 | bitri 183 | . . . . . . 7 |
21 | df-rab 2425 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2206 | . . . . . . . 8 |
23 | df-clab 2126 | . . . . . . . . 9 | |
24 | sban 1928 | . . . . . . . . 9 | |
25 | 23, 24 | bitri 183 | . . . . . . . 8 |
26 | 22, 25 | bitri 183 | . . . . . . 7 |
27 | 17, 20, 26 | 3imtr4i 200 | . . . . . 6 |
28 | elex2 2702 | . . . . . 6 | |
29 | 27, 28 | syl 14 | . . . . 5 |
30 | rabn0m 3390 | . . . . 5 | |
31 | 29, 30 | sylib 121 | . . . 4 |
32 | 6 | snex 4109 | . . . . 5 |
33 | sseq1 3120 | . . . . . 6 | |
34 | rexeq 2627 | . . . . . 6 | |
35 | 33, 34 | anbi12d 464 | . . . . 5 |
36 | 32, 35 | spcev 2780 | . . . 4 |
37 | 11, 31, 36 | syl2anc 408 | . . 3 |
38 | 37 | exlimiv 1577 | . 2 |
39 | 5, 38 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wsb 1735 cab 2125 wrex 2417 crab 2420 wss 3071 csn 3527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-rab 2425 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 |
This theorem is referenced by: (None) |
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