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Mirrors > Home > MPE Home > Th. List > exss | Structured version Visualization version 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 | df-rab 2921 | . . . 4 | |
2 | 1 | neeq1i 2858 | . . 3 |
3 | rabn0 3958 | . . 3 | |
4 | n0 3931 | . . 3 | |
5 | 2, 3, 4 | 3bitr3i 290 | . 2 |
6 | vex 3203 | . . . . . 6 | |
7 | 6 | snss 4316 | . . . . 5 |
8 | ssab2 3686 | . . . . . 6 | |
9 | sstr2 3610 | . . . . . 6 | |
10 | 8, 9 | mpi 20 | . . . . 5 |
11 | 7, 10 | sylbi 207 | . . . 4 |
12 | simpr 477 | . . . . . . . 8 | |
13 | equsb1 2368 | . . . . . . . . 9 | |
14 | velsn 4193 | . . . . . . . . . 10 | |
15 | 14 | sbbii 1887 | . . . . . . . . 9 |
16 | 13, 15 | mpbir 221 | . . . . . . . 8 |
17 | 12, 16 | jctil 560 | . . . . . . 7 |
18 | df-clab 2609 | . . . . . . . 8 | |
19 | sban 2399 | . . . . . . . 8 | |
20 | 18, 19 | bitri 264 | . . . . . . 7 |
21 | df-rab 2921 | . . . . . . . . 9 | |
22 | 21 | eleq2i 2693 | . . . . . . . 8 |
23 | df-clab 2609 | . . . . . . . . 9 | |
24 | sban 2399 | . . . . . . . . 9 | |
25 | 23, 24 | bitri 264 | . . . . . . . 8 |
26 | 22, 25 | bitri 264 | . . . . . . 7 |
27 | 17, 20, 26 | 3imtr4i 281 | . . . . . 6 |
28 | ne0i 3921 | . . . . . 6 | |
29 | 27, 28 | syl 17 | . . . . 5 |
30 | rabn0 3958 | . . . . 5 | |
31 | 29, 30 | sylib 208 | . . . 4 |
32 | snex 4908 | . . . . 5 | |
33 | sseq1 3626 | . . . . . 6 | |
34 | rexeq 3139 | . . . . . 6 | |
35 | 33, 34 | anbi12d 747 | . . . . 5 |
36 | 32, 35 | spcev 3300 | . . . 4 |
37 | 11, 31, 36 | syl2anc 693 | . . 3 |
38 | 37 | exlimiv 1858 | . 2 |
39 | 5, 38 | sylbi 207 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wex 1704 wsb 1880 wcel 1990 cab 2608 wne 2794 wrex 2913 crab 2916 wss 3574 c0 3915 csn 4177 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: (None) |
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