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Mirrors > Home > ILE Home > Th. List > eqsnm | GIF version |
Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
eqsnm | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssnm 3769 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵})) | |
2 | dfss3 3160 | . . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵}) | |
3 | velsn 3624 | . . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 3 | ralbii 2496 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
5 | 2, 4 | bitri 184 | . 2 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵) |
6 | 1, 5 | bitr3di 195 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ⊆ wss 3144 {csn 3607 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-in 3150 df-ss 3157 df-sn 3613 |
This theorem is referenced by: 01eq0ring 13529 nninfall 15196 |
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