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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 4688 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
2 | rabsnif.f | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 3784 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 3 | ifbid 4514 | . . 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 4347 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ | |
7 | ifid 4531 | . . . 4 ⊢ if(𝜓, ∅, ∅) = ∅ | |
8 | 6, 7 | eqtr4i 2768 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅) |
9 | snprc 4683 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | rabeqdv 3425 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑}) |
12 | 10 | ifeq1d 4510 | . . 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 205 = wceq 1542 ∈ wcel 2107 {crab 3410 Vcvv 3448 [wsbc 3744 ∅c0 4287 ifcif 4491 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-nul 4288 df-if 4492 df-sn 4592 |
This theorem is referenced by: suppsnop 8114 m1detdiag 21962 left1s 27246 right1s 27247 1loopgrvd2 28493 1hevtxdg1 28496 1egrvtxdg1 28499 |
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