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Mirrors > Home > MPE Home > Th. List > Mathboxes > truniALT | Structured version Visualization version Unicode version |
Description: The union of a class of transitive sets is transitive. Alternate proof of truni 4767. truniALT 38751 is truniALTVD 39114 without virtual deductions and was automatically derived from truniALTVD 39114 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
truniALT |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . . . 6 | |
2 | 1 | a1i 11 | . . . . 5 |
3 | eluni 4439 | . . . . 5 | |
4 | 2, 3 | syl6ib 241 | . . . 4 |
5 | simpl 473 | . . . . . . . . 9 | |
6 | 5 | a1i 11 | . . . . . . . 8 |
7 | simpl 473 | . . . . . . . . 9 | |
8 | 7 | 2a1i 12 | . . . . . . . 8 |
9 | simpr 477 | . . . . . . . . . 10 | |
10 | 9 | 2a1i 12 | . . . . . . . . 9 |
11 | rspsbc 3518 | . . . . . . . . . . 11 | |
12 | 11 | com12 32 | . . . . . . . . . 10 |
13 | 10, 12 | syl6d 75 | . . . . . . . . 9 |
14 | trsbc 38750 | . . . . . . . . . 10 | |
15 | 14 | biimpd 219 | . . . . . . . . 9 |
16 | 10, 13, 15 | ee33 38727 | . . . . . . . 8 |
17 | trel 4759 | . . . . . . . . 9 | |
18 | 17 | expdcom 455 | . . . . . . . 8 |
19 | 6, 8, 16, 18 | ee233 38725 | . . . . . . 7 |
20 | elunii 4441 | . . . . . . . 8 | |
21 | 20 | ex 450 | . . . . . . 7 |
22 | 19, 10, 21 | ee33 38727 | . . . . . 6 |
23 | 22 | alrimdv 1857 | . . . . 5 |
24 | 19.23v 1902 | . . . . 5 | |
25 | 23, 24 | syl6ib 241 | . . . 4 |
26 | 4, 25 | mpdd 43 | . . 3 |
27 | 26 | alrimivv 1856 | . 2 |
28 | dftr2 4754 | . 2 | |
29 | 27, 28 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wex 1704 wcel 1990 wral 2912 wsbc 3435 cuni 4436 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-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-tr 4753 |
This theorem is referenced by: (None) |
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