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Mirrors > Home > ILE Home > Th. List > enq0ex | Unicode version |
Description: The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Ref | Expression |
---|---|
enq0ex | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4577 | . . . 4 | |
2 | niex 7274 | . . . 4 | |
3 | 1, 2 | xpex 4726 | . . 3 |
4 | 3, 3 | xpex 4726 | . 2 |
5 | df-enq0 7386 | . . 3 ~Q0 | |
6 | opabssxp 4685 | . . 3 | |
7 | 5, 6 | eqsstri 3179 | . 2 ~Q0 |
8 | 4, 7 | ssexi 4127 | 1 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 wcel 2141 cvv 2730 cop 3586 copab 4049 com 4574 cxp 4609 (class class class)co 5853 comu 6393 cnpi 7234 ~Q0 ceq0 7248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-opab 4051 df-iom 4575 df-xp 4617 df-ni 7266 df-enq0 7386 |
This theorem is referenced by: nqnq0 7403 addnnnq0 7411 mulnnnq0 7412 addclnq0 7413 mulclnq0 7414 prarloclemcalc 7464 |
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