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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 4650 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
2 | rabsnif.f | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 3807 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 3 | ifbid 4485 | . . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅)) |
5 | 1, 4 | syl5eq 2865 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
6 | rab0 4334 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ | |
7 | ifid 4502 | . . . 4 ⊢ if(𝜓, ∅, ∅) = ∅ | |
8 | 6, 7 | eqtr4i 2844 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅) |
9 | snprc 4645 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 217 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | rabeqdv 3482 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑}) |
12 | 10 | ifeq1d 4481 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅)) |
13 | 8, 11, 12 | 3eqtr4a 2879 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
14 | 5, 13 | pm2.61i 183 | 1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 [wsbc 3769 ∅c0 4288 ifcif 4463 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-nul 4289 df-if 4464 df-sn 4558 |
This theorem is referenced by: suppsnop 7833 m1detdiag 21134 1loopgrvd2 27212 1hevtxdg1 27215 1egrvtxdg1 27218 |
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