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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 3547 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | elex 2566 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | snexg 3936 | . . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
5 | 4 | adantr 261 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴} ∈ V) |
6 | elex 2566 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
7 | prexg 3947 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
8 | 2, 6, 7 | syl2an 273 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
9 | prexg 3947 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} ∈ V) | |
10 | 5, 8, 9 | syl2anc 391 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {{𝐴}, {𝐴, 𝐵}} ∈ V) |
11 | 1, 10 | eqeltrd 2114 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 Vcvv 2557 {csn 3375 {cpr 3376 〈cop 3378 |
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-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 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-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: opex 3966 otexg 3967 fliftel1 5434 oprabid 5537 ovexg 5539 eloprabga 5591 op1st 5773 op2nd 5774 ot1stg 5779 ot2ndg 5780 ot3rdgg 5781 elxp6 5796 mpt2fvex 5829 algrflem 5850 algrflemg 5851 mpt2xopoveq 5855 brtposg 5869 tfr0 5937 tfrlemisucaccv 5939 tfrlemibxssdm 5941 tfrlemibfn 5942 tfrlemi14d 5947 mulpipq2 6469 enq0breq 6534 addvalex 6920 peano2nnnn 6929 axcnre 6955 frec2uzrdg 9195 frecuzrdg0 9200 frecuzrdgsuc 9201 |
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