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Mirrors > Home > ILE Home > Th. List > prmg | GIF version |
Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
prmg | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmg 3679 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴}) | |
2 | orc 702 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | velsn 3578 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | vex 2715 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 4 | elpr 3582 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
6 | 2, 3, 5 | 3imtr4i 200 | . . 3 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵}) |
7 | 6 | eximi 1580 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴, 𝐵}) |
8 | 1, 7 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1335 ∃wex 1472 ∈ wcel 2128 {csn 3561 {cpr 3562 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3567 df-pr 3568 |
This theorem is referenced by: prm 3684 opm 4196 onintexmid 4534 |
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