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Mirrors > Home > MPE Home > Th. List > elopOLD | Structured version Visualization version GIF version |
Description: Obsolete version of elop 4935, with one extraneous hypothesis. Obsolete as of 25-Dec-2020 . (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elop.1 | ⊢ 𝐵 ∈ V |
elop.2 | ⊢ 𝐶 ∈ V |
elopOLD.3 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elopOLD | ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elop.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | elop.2 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | dfop 4401 | . . 3 ⊢ 〈𝐵, 𝐶〉 = {{𝐵}, {𝐵, 𝐶}} |
4 | 3 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}}) |
5 | elopOLD.3 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | elpr 4198 | . 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
7 | 4, 6 | bitri 264 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 {cpr 4179 〈cop 4183 |
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-3an 1039 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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: (None) |
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