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Mirrors > Home > ILE Home > Th. List > Mathboxes > if0ab | GIF version |
Description: Expression of a
conditional class as a class abstraction when the False
alternative is the empty class: in that case, the conditional class is
the extension, in the True alternative, of the condition.
Remark: a consequence which could be formalized is the inclusion ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3582, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 14180 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.) |
Ref | Expression |
---|---|
if0ab | ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3426 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 930 | . . . . 5 ⊢ ¬ (𝑥 ∈ ∅ ∧ ¬ 𝜑) |
3 | 2 | biorfi 746 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))) |
4 | 3 | bicomi 132 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | abbii 2293 | . 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
6 | df-if 3535 | . 2 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} | |
7 | df-rab 2464 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
8 | 5, 6, 7 | 3eqtr4i 2208 | 1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 {cab 2163 {crab 2459 ∅c0 3422 ifcif 3534 |
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-in1 614 ax-in2 615 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-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2739 df-dif 3131 df-nul 3423 df-if 3535 |
This theorem is referenced by: (None) |
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