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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 3722 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴}) | |
2 | orc 713 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | velsn 3621 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | vex 2752 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 4 | elpr 3625 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
6 | 2, 3, 5 | 3imtr4i 201 | . . 3 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵}) |
7 | 6 | eximi 1610 | . 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 709 = wceq 1363 ∃wex 1502 ∈ wcel 2158 {csn 3604 {cpr 3605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 |
This theorem is referenced by: prm 3727 opm 4246 onintexmid 4584 subrngin 13490 subrgin 13521 lssincl 13631 |
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