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Mirrors > Home > MPE Home > Th. List > grupr | Structured version Visualization version Unicode version |
Description: A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
grupr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elgrug 9614 | . . . . . . 7 | |
2 | 1 | ibi 256 | . . . . . 6 |
3 | 2 | simprd 479 | . . . . 5 |
4 | preq2 4269 | . . . . . . . . . 10 | |
5 | 4 | eleq1d 2686 | . . . . . . . . 9 |
6 | 5 | rspccv 3306 | . . . . . . . 8 |
7 | 6 | 3ad2ant2 1083 | . . . . . . 7 |
8 | 7 | com12 32 | . . . . . 6 |
9 | 8 | ralimdv 2963 | . . . . 5 |
10 | 3, 9 | syl5com 31 | . . . 4 |
11 | preq1 4268 | . . . . . 6 | |
12 | 11 | eleq1d 2686 | . . . . 5 |
13 | 12 | rspccv 3306 | . . . 4 |
14 | 10, 13 | syl6 35 | . . 3 |
15 | 14 | com23 86 | . 2 |
16 | 15 | 3imp 1256 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cpw 4158 cpr 4179 cuni 4436 wtr 4752 crn 5115 (class class class)co 6650 cmap 7857 cgru 9612 |
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-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-tr 4753 df-iota 5851 df-fv 5896 df-ov 6653 df-gru 9613 |
This theorem is referenced by: grusn 9626 gruop 9627 gruun 9628 gruwun 9635 intgru 9636 |
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