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Mirrors > Home > ILE Home > Th. List > rabsnt | GIF version |
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabsnt.1 | ⊢ 𝐵 ∈ V |
rabsnt.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabsnt | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsnt.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | 1 | snid 3556 | . . 3 ⊢ 𝐵 ∈ {𝐵} |
3 | id 19 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵}) | |
4 | 2, 3 | eleqtrrid 2229 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
5 | rabsnt.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
6 | 5 | elrab 2840 | . . 3 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ∈ 𝐴 ∧ 𝜓)) |
7 | 6 | simprbi 273 | . 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜓) |
8 | 4, 7 | syl 14 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {crab 2420 Vcvv 2686 {csn 3527 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-sn 3533 |
This theorem is referenced by: ontr2exmid 4440 onsucsssucexmid 4442 ordsoexmid 4477 unfiexmid 6806 |
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