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Mirrors > Home > ILE Home > Th. List > eusv2 | GIF version |
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
eusv2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eusv2 | ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusv2.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | eusv2nf 4452 | . 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
3 | eusvnfb 4450 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) | |
4 | 1, 3 | mpbiran2 941 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
5 | 2, 4 | bitr4i 187 | 1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∃!weu 2026 ∈ wcel 2148 Ⅎwnfc 2306 Vcvv 2737 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-sn 3597 df-pr 3598 df-uni 3808 |
This theorem is referenced by: (None) |
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