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Mirrors > Home > MPE Home > Th. List > clel4g | Structured version Visualization version GIF version |
Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2175. (Revised by BJ, 1-Sep-2024.) |
Ref | Expression |
---|---|
clel4g | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2821 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵) | |
2 | biimt 360 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵))) |
4 | 19.23v 1940 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵)) | |
5 | 3, 4 | bitr4di 289 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵))) |
6 | eleq2 2828 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
7 | 6 | bicomd 223 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥)) |
8 | 7 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
9 | 8 | albii 1816 | . 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
10 | 5, 9 | bitrdi 287 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: clel4 3664 intprg 4986 |
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