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Mirrors > Home > ILE Home > Th. List > tpexg | GIF version |
Description: An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
Ref | Expression |
---|---|
tpexg | ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {𝐴, 𝐵, 𝐶} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3474 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prexg 4062 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ V) | |
3 | snexg 4040 | . . . . 5 ⊢ (𝐶 ∈ 𝑊 → {𝐶} ∈ V) | |
4 | 2, 3 | anim12i 332 | . . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → ({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V)) |
5 | 4 | 3impa 1141 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V)) |
6 | unexg 4293 | . . 3 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ V) | |
7 | 5, 6 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({𝐴, 𝐵} ∪ {𝐶}) ∈ V) |
8 | 1, 7 | syl5eqel 2181 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {𝐴, 𝐵, 𝐶} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 927 ∈ wcel 1445 Vcvv 2633 ∪ cun 3011 {csn 3466 {cpr 3467 {ctp 3468 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-tp 3474 df-uni 3676 |
This theorem is referenced by: (None) |
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