Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4257 | . . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) | |
2 | 1 | eqeq2i 2634 | . 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)) |
3 | ifeqor 4132 | . . . 4 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅) | |
4 | orcom 402 | . . . 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 220 | . . 3 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}) |
6 | eqeq1 2626 | . . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅)) | |
7 | eqeq1 2626 | . . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = {𝐴} ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴})) | |
8 | 6, 7 | orbi12d 746 | . . 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 248 | . 2 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ∨ 𝑀 = {𝐴})) |
10 | 2, 9 | sylbi 207 | 1 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 = wceq 1483 {crab 2916 [wsbc 3435 ∅c0 3915 ifcif 4086 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-nul 3916 df-if 4087 df-sn 4178 |
This theorem is referenced by: hashrabrsn 13161 hashrabsn01 13162 hashrabsn1 13163 dvnprodlem3 40163 |
Copyright terms: Public domain | W3C validator |