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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 4618 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
2 | rabsnif.f | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 3758 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 3 | ifbid 4447 | . . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅)) |
5 | 1, 4 | syl5eq 2845 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
6 | rab0 4291 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ | |
7 | ifid 4464 | . . . 4 ⊢ if(𝜓, ∅, ∅) = ∅ | |
8 | 6, 7 | eqtr4i 2824 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅) |
9 | snprc 4613 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 219 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | rabeqdv 3432 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑}) |
12 | 10 | ifeq1d 4443 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅)) |
13 | 8, 11, 12 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)) |
14 | 5, 13 | pm2.61i 185 | 1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 [wsbc 3720 ∅c0 4243 ifcif 4425 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-nul 4244 df-if 4426 df-sn 4526 |
This theorem is referenced by: suppsnop 7827 m1detdiag 21202 1loopgrvd2 27293 1hevtxdg1 27296 1egrvtxdg1 27299 |
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