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| Mirrors > Home > MPE Home > Th. List > rabsnif | Structured version Visualization version GIF version | ||
| Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.) |
| Ref | Expression |
|---|---|
| rabsnif.f | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rabsnif | ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabsnifsb 4722 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
| 2 | rabsnif.f | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | sbcieg 3828 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
| 4 | 3 | ifbid 4549 | . . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅)) |
| 5 | 1, 4 | eqtrid 2789 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
| 6 | rab0 4386 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ | |
| 7 | ifid 4566 | . . . 4 ⊢ if(𝜓, ∅, ∅) = ∅ | |
| 8 | 6, 7 | eqtr4i 2768 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅) |
| 9 | snprc 4717 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 10 | 9 | biimpi 216 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 11 | 10 | rabeqdv 3452 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑}) |
| 12 | 10 | ifeq1d 4545 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅)) |
| 13 | 8, 11, 12 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
| 14 | 5, 13 | pm2.61i 182 | 1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 [wsbc 3788 ∅c0 4333 ifcif 4525 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-nul 4334 df-if 4526 df-sn 4627 |
| This theorem is referenced by: suppsnop 8203 m1detdiag 22603 left1s 27933 right1s 27934 1loopgrvd2 29521 1hevtxdg1 29524 1egrvtxdg1 29527 |
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