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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 3635 | . . 3 ⊢ 𝐵 ∈ {𝐵} |
3 | id 19 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵}) | |
4 | 2, 3 | eleqtrrid 2277 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
5 | rabsnt.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
6 | 5 | elrab 2905 | . . 3 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ∈ 𝐴 ∧ 𝜓)) |
7 | 6 | simprbi 275 | . 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 105 = wceq 1363 ∈ wcel 2158 {crab 2469 Vcvv 2749 {csn 3604 |
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-rab 2474 df-v 2751 df-sn 3610 |
This theorem is referenced by: ontr2exmid 4536 onsucsssucexmid 4538 ordsoexmid 4573 unfiexmid 6931 |
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