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Mirrors > Home > MPE Home > Th. List > rabrsn | 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 Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.) |
Ref | Expression |
---|---|
rabrsn | ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsnifsb 4725 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
2 | 1 | eqeq2i 2745 | . 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)) |
3 | ifeqor 4578 | . . . 4 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅) | |
4 | orcom 868 | . . . 4 ⊢ ((if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅) ↔ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴})) | |
5 | 3, 4 | mpbi 229 | . . 3 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}) |
6 | eqeq1 2736 | . . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅)) | |
7 | eqeq1 2736 | . . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = {𝐴} ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴})) | |
8 | 6, 7 | orbi12d 917 | . . 3 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}))) |
9 | 5, 8 | mpbiri 257 | . 2 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ∨ 𝑀 = {𝐴})) |
10 | 2, 9 | sylbi 216 | 1 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1541 {crab 3432 [wsbc 3776 ∅c0 4321 ifcif 4527 {csn 4627 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-rab 3433 df-sbc 3777 df-dif 3950 df-nul 4322 df-if 4528 df-sn 4628 |
This theorem is referenced by: hashrabrsn 14328 hashrabsn01 14329 hashrabsn1 14330 dvnprodlem3 44650 |
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