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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elisset | Structured version Visualization version GIF version |
Description: Remove from elisset 3215 dependency on ax-ext 2602 (and on df-cleq 2615 and df-v 3202). This proof uses only df-clab 2609 and df-clel 2618 on top of first-order logic. It only requires ax-1--7 and sp 2053. Use bj-elissetv 32861 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-elisset | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-elissetv 32861 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴) | |
2 | bj-denotes 32858 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylib 208 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∃wex 1704 ∈ wcel 1990 |
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-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 df-clab 2609 df-clel 2618 |
This theorem is referenced by: bj-isseti 32864 bj-ceqsalt 32875 bj-ceqsalg 32878 bj-spcimdv 32884 bj-vtoclg1f 32911 |
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