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Mirrors > Home > ILE Home > Th. List > prexgOLD | 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 3480, prprc1 3478, and prprc2 3479. This is a special case of prexg 3947 and new proofs should use prexg 3947 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3947 and then remove it. |
Ref | Expression |
---|---|
prexgOLD |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3448 |
. . . . . 6
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2 | 1 | eleq1d 2106 |
. . . . 5
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3 | zfpair2 3945 |
. . . . 5
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4 | 2, 3 | vtoclg 2613 |
. . . 4
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5 | preq1 3447 |
. . . . 5
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6 | 5 | eleq1d 2106 |
. . . 4
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7 | 4, 6 | syl5ib 143 |
. . 3
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8 | 7 | vtocleg 2624 |
. 2
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9 | 8 | imp 115 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: prelpwi 3950 opexgOLD 3965 opi2 3970 opth 3974 opeqsn 3989 opeqpr 3990 uniop 3992 unex 4176 op1stb 4209 op1stbg 4210 opthreg 4280 relop 4486 |
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