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Mirrors > Home > ILE Home > Th. List > xpex | GIF version |
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
xpex.1 | ⊢ 𝐴 ∈ V |
xpex.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
xpex | ⊢ (𝐴 × 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | xpex.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | xpexg 4736 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 × 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 × cxp 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-opab 4062 df-xp 4628 |
This theorem is referenced by: oprabex 6122 oprabex3 6123 mpoexw 6207 fnpm 6649 mapsnf1o2 6689 xpsnen 6814 endisj 6817 xpcomen 6820 xpassen 6823 xpmapenlem 6842 0ct 7099 exmidomni 7133 exmidfodomrlemim 7193 enqex 7337 nqex 7340 enq0ex 7416 nq0ex 7417 npex 7450 enrex 7714 addvalex 7821 axcnex 7836 ixxex 9873 fxnn0nninf 10411 inftonninf 10414 shftfval 10801 qnumval 12155 qdenval 12156 qnnen 12402 txuni2 13389 txbas 13391 eltx 13392 txcnp 13404 txcnmpt 13406 txrest 13409 txlm 13412 reldvg 13781 |
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