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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 3578 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prexg 4183 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ∈ V) | |
3 | snexg 4157 | . . . . 5 ⊢ (𝐶 ∈ 𝑊 → {𝐶} ∈ V) | |
4 | 2, 3 | anim12i 336 | . . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → ({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V)) |
5 | 4 | 3impa 1183 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V)) |
6 | unexg 4415 | . . 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 | eqeltrid 2251 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {𝐴, 𝐵, 𝐶} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 {csn 3570 {cpr 3571 {ctp 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-uni 3784 |
This theorem is referenced by: (None) |
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