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Mirrors > Home > MPE Home > Th. List > Mathboxes > trintALTVD | Structured version Visualization version Unicode version |
Description: The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 39117.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trintALT 39117 is trintALTVD 39116 without virtual deductions and was
automatically derived from trintALTVD 39116.
|
Ref | Expression |
---|---|
trintALTVD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn2 38838 | . . . . . . 7 | |
2 | simpl 473 | . . . . . . 7 | |
3 | 1, 2 | e2 38856 | . . . . . 6 |
4 | idn3 38840 | . . . . . . . . . . 11 | |
5 | idn1 38790 | . . . . . . . . . . . 12 | |
6 | rspsbc 3518 | . . . . . . . . . . . 12 | |
7 | 4, 5, 6 | e31 38978 | . . . . . . . . . . 11 |
8 | trsbc 38750 | . . . . . . . . . . . 12 | |
9 | 8 | biimpd 219 | . . . . . . . . . . 11 |
10 | 4, 7, 9 | e33 38961 | . . . . . . . . . 10 |
11 | simpr 477 | . . . . . . . . . . . . . 14 | |
12 | 1, 11 | e2 38856 | . . . . . . . . . . . . 13 |
13 | elintg 4483 | . . . . . . . . . . . . . 14 | |
14 | 13 | ibi 256 | . . . . . . . . . . . . 13 |
15 | 12, 14 | e2 38856 | . . . . . . . . . . . 12 |
16 | rsp 2929 | . . . . . . . . . . . 12 | |
17 | 15, 16 | e2 38856 | . . . . . . . . . . 11 |
18 | pm2.27 42 | . . . . . . . . . . 11 | |
19 | 4, 17, 18 | e32 38985 | . . . . . . . . . 10 |
20 | trel 4759 | . . . . . . . . . . 11 | |
21 | 20 | expd 452 | . . . . . . . . . 10 |
22 | 10, 3, 19, 21 | e323 38993 | . . . . . . . . 9 |
23 | 22 | in3 38834 | . . . . . . . 8 |
24 | 23 | gen21 38844 | . . . . . . 7 |
25 | df-ral 2917 | . . . . . . . 8 | |
26 | 25 | biimpri 218 | . . . . . . 7 |
27 | 24, 26 | e2 38856 | . . . . . 6 |
28 | elintg 4483 | . . . . . . 7 | |
29 | 28 | biimprd 238 | . . . . . 6 |
30 | 3, 27, 29 | e22 38896 | . . . . 5 |
31 | 30 | in2 38830 | . . . 4 |
32 | 31 | gen12 38843 | . . 3 |
33 | dftr2 4754 | . . . 4 | |
34 | 33 | biimpri 218 | . . 3 |
35 | 32, 34 | e1a 38852 | . 2 |
36 | 35 | in1 38787 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wcel 1990 wral 2912 wsbc 3435 cint 4475 wtr 4752 |
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 |
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-ral 2917 df-v 3202 df-sbc 3436 df-in 3581 df-ss 3588 df-uni 4437 df-int 4476 df-tr 4753 df-vd1 38786 df-vd2 38794 df-vd3 38806 |
This theorem is referenced by: (None) |
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