![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > rabxp | Unicode version |
Description: Membership in a class builder restricted to a Cartesian product. (Contributed by NM, 20-Feb-2014.) |
Ref | Expression |
---|---|
rabxp.1 |
Ref | Expression |
---|---|
rabxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 5021 | . . . . 5 | |
2 | 1 | anbi1i 695 | . . . 4 |
3 | 19.41vv 1772 | . . . 4 | |
4 | anass 649 | . . . . . 6 | |
5 | rabxp.1 | . . . . . . . . 9 | |
6 | 5 | anbi2d 703 | . . . . . . . 8 |
7 | df-3an 975 | . . . . . . . 8 | |
8 | 6, 7 | syl6bbr 263 | . . . . . . 7 |
9 | 8 | pm5.32i 637 | . . . . . 6 |
10 | 4, 9 | bitri 249 | . . . . 5 |
11 | 10 | 2exbii 1668 | . . . 4 |
12 | 2, 3, 11 | 3bitr2i 273 | . . 3 |
13 | 12 | abbii 2591 | . 2 |
14 | df-rab 2816 | . 2 | |
15 | df-opab 4511 | . 2 | |
16 | 13, 14, 15 | 3eqtr4i 2496 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 /\ w3a 973 = wceq 1395
E. wex 1612 e. wcel 1818 { cab 2442
{ crab 2811 <. cop 4035 { copab 4509 X. cxp 5002 |
This theorem is referenced by: fgraphxp 31171 cicer 32590 dib1dim 36892 diclspsn 36921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 df-opab 4511 df-xp 5010 |
Copyright terms: Public domain | W3C validator |