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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 3360 | . . 3 | |
2 | df-rab 2402 | . . . . 5 | |
3 | 2 | eleq2i 2184 | . . . 4 |
4 | 3 | exbii 1569 | . . 3 |
5 | 1, 4 | bitr3i 185 | . 2 |
6 | vex 2663 | . . . . . 6 | |
7 | 6 | snss 3619 | . . . . 5 |
8 | ssab2 3151 | . . . . . 6 | |
9 | sstr2 3074 | . . . . . 6 | |
10 | 8, 9 | mpi 15 | . . . . 5 |
11 | 7, 10 | sylbi 120 | . . . 4 |
12 | simpr 109 | . . . . . . . 8 | |
13 | equsb1 1743 | . . . . . . . . 9 | |
14 | velsn 3514 | . . . . . . . . . 10 | |
15 | 14 | sbbii 1723 | . . . . . . . . 9 |
16 | 13, 15 | mpbir 145 | . . . . . . . 8 |
17 | 12, 16 | jctil 310 | . . . . . . 7 |
18 | df-clab 2104 | . . . . . . . 8 | |
19 | sban 1906 | . . . . . . . 8 | |
20 | 18, 19 | bitri 183 | . . . . . . 7 |
21 | df-rab 2402 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2184 | . . . . . . . 8 |
23 | df-clab 2104 | . . . . . . . . 9 | |
24 | sban 1906 | . . . . . . . . 9 | |
25 | 23, 24 | bitri 183 | . . . . . . . 8 |
26 | 22, 25 | bitri 183 | . . . . . . 7 |
27 | 17, 20, 26 | 3imtr4i 200 | . . . . . 6 |
28 | elex2 2676 | . . . . . 6 | |
29 | 27, 28 | syl 14 | . . . . 5 |
30 | rabn0m 3360 | . . . . 5 | |
31 | 29, 30 | sylib 121 | . . . 4 |
32 | 6 | snex 4079 | . . . . 5 |
33 | sseq1 3090 | . . . . . 6 | |
34 | rexeq 2604 | . . . . . 6 | |
35 | 33, 34 | anbi12d 464 | . . . . 5 |
36 | 32, 35 | spcev 2754 | . . . 4 |
37 | 11, 31, 36 | syl2anc 408 | . . 3 |
38 | 37 | exlimiv 1562 | . 2 |
39 | 5, 38 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 wsb 1720 cab 2103 wrex 2394 crab 2397 wss 3041 csn 3497 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-rab 2402 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 |
This theorem is referenced by: (None) |
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