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Mirrors > Home > ILE Home > Th. List > opexg | GIF version |
Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Ref | Expression |
---|---|
opexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3626 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | elex 2631 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | snexg 4025 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
5 | 4 | adantr 271 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴} ∈ V) |
6 | elex 2631 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
7 | prexg 4047 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
8 | 2, 6, 7 | syl2an 284 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
9 | prexg 4047 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} ∈ V) | |
10 | 5, 8, 9 | syl2anc 404 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {{𝐴}, {𝐴, 𝐵}} ∈ V) |
11 | 1, 10 | eqeltrd 2165 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1439 Vcvv 2620 {csn 3450 {cpr 3451 〈cop 3453 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 |
This theorem is referenced by: opex 4065 otexg 4066 opeliunxp 4506 opbrop 4530 relsnopg 4555 opswapg 4930 elxp4 4931 elxp5 4932 resfunexg 5532 fliftel 5586 fliftel1 5587 oprabid 5695 ovexg 5697 eloprabga 5749 op1st 5931 op2nd 5932 ot1stg 5937 ot2ndg 5938 ot3rdgg 5939 elxp6 5954 mpt2fvex 5987 algrflem 6008 algrflemg 6009 mpt2xopoveq 6019 brtposg 6033 tfr0dm 6101 tfrlemisucaccv 6104 tfrlemibxssdm 6106 tfrlemibfn 6107 tfrlemi14d 6112 tfr1onlemsucaccv 6120 tfr1onlembxssdm 6122 tfr1onlembfn 6123 tfr1onlemres 6128 tfrcllemsucaccv 6133 tfrcllembxssdm 6135 tfrcllembfn 6136 tfrcllemres 6141 fnfi 6700 djulclb 6801 djur 6811 1stinl 6819 2ndinl 6820 1stinr 6821 2ndinr 6822 mulpipq2 6991 enq0breq 7056 addvalex 7442 peano2nnnn 7451 axcnre 7477 frec2uzrdg 9877 frecuzrdg0 9881 frecuzrdgg 9884 frecuzrdg0t 9890 zfz1isolem1 10306 eucalgval2 11374 crth 11539 phimullem 11540 setsvala 11586 setsex 11587 setsfun 11590 setsfun0 11591 setsresg 11593 setscom 11595 strslfv 11599 setsslid 11605 strle1g 11645 1strbas 11654 2strbasg 11656 2stropg 11657 2strbas1g 11659 2strop1g 11660 rngbaseg 11671 rngplusgg 11672 rngmulrg 11673 srngbased 11678 srngplusgd 11679 srngmulrd 11680 srnginvld 11681 lmodbased 11689 lmodplusgd 11690 lmodscad 11691 lmodvscad 11692 ipsbased 11697 ipsaddgd 11698 ipsmulrd 11699 ipsscad 11700 ipsvscad 11701 ipsipd 11702 topgrpbasd 11707 topgrpplusgd 11708 topgrptsetd 11709 |
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