Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > preqsnOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of preqsn 4393 as of 23-Jul-2021. (Contributed by NM, 3-Jun-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
preqsn.1 | ⊢ 𝐴 ∈ V |
preqsn.2 | ⊢ 𝐵 ∈ V |
preqsn.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
preqsnOLD | ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4190 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | eqeq2i 2634 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶}) |
3 | preqsn.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | preqsn.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | preqsn.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | 3, 4, 5, 5 | preq12b 4382 | . . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
7 | oridm 536 | . . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) | |
8 | eqtr3 2643 | . . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | |
9 | simpr 477 | . . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
10 | 8, 9 | jca 554 | . . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
11 | eqtr 2641 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | |
12 | simpr 477 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
13 | 11, 12 | jca 554 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
14 | 10, 13 | impbii 199 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
15 | 7, 14 | bitri 264 | . . 3 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
16 | 6, 15 | bitri 264 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
17 | 2, 16 | bitri 264 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 {cpr 4179 |
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-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |