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Mirrors > Home > MPE Home > Th. List > pr1eqbg | Structured version Visualization version Unicode version |
Description: A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
Ref | Expression |
---|---|
pr1eqbg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 | |
2 | 1 | biantru 526 | . . . 4 |
3 | 2 | orbi2i 541 | . . 3 |
4 | 3 | a1i 11 | . 2 |
5 | neneq 2800 | . . . . 5 | |
6 | 5 | adantl 482 | . . . 4 |
7 | 6 | intnanrd 963 | . . 3 |
8 | biorf 420 | . . 3 | |
9 | 7, 8 | syl 17 | . 2 |
10 | 3simpa 1058 | . . . . 5 | |
11 | 3simpc 1060 | . . . . 5 | |
12 | 10, 11 | jca 554 | . . . 4 |
13 | 12 | adantr 481 | . . 3 |
14 | preq12bg 4386 | . . 3 | |
15 | 13, 14 | syl 17 | . 2 |
16 | 4, 9, 15 | 3bitr4d 300 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cpr 4179 |
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-ne 2795 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: pr1nebg 4391 |
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