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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 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 and then remove it. |
Ref | Expression |
---|---|
prexgOLD |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3439 |
. . . . . 6
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2 | 1 | eleq1d 2103 |
. . . . 5
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3 | zfpair2 3936 |
. . . . 5
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4 | 2, 3 | vtoclg 2607 |
. . . 4
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5 | preq1 3438 |
. . . . 5
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6 | 5 | eleq1d 2103 |
. . . 4
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7 | 4, 6 | syl5ib 143 |
. . 3
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8 | 7 | vtocleg 2618 |
. 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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: prelpwi 3941 opexgOLD 3956 opi2 3961 opth 3965 opeqsn 3980 opeqpr 3981 uniop 3983 unex 4142 op1stb 4175 op1stbg 4176 opthreg 4234 relop 4429 |
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