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Mirrors > Home > ILE Home > Th. List > prexg | Unicode version |
Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3502, prprc1 3500, and prprc2 3501. (Contributed by Jim Kingdon, 16-Sep-2018.) |
Ref | Expression |
---|---|
prexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3470 | . . . . . 6 | |
2 | 1 | eleq1d 2147 | . . . . 5 |
3 | zfpair2 3965 | . . . . 5 | |
4 | 2, 3 | vtoclg 2658 | . . . 4 |
5 | preq1 3469 | . . . . 5 | |
6 | 5 | eleq1d 2147 | . . . 4 |
7 | 4, 6 | syl5ib 152 | . . 3 |
8 | 7 | vtocleg 2669 | . 2 |
9 | 8 | imp 122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: prelpwi 3969 opexg 3983 opi2 3988 opth 3992 opeqsn 4007 opeqpr 4008 uniop 4010 unex 4194 tpexg 4197 op1stb 4227 op1stbg 4228 onun2 4234 opthreg 4299 relop 4504 acexmidlemv 5530 pr2ne 6461 xrex 8910 |
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