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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 2833 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵) | |
2 | biimt 364 | . . . 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 1943 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵)) | |
5 | 3, 4 | bitr4di 292 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵))) |
6 | eleq2 2840 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
7 | 6 | bicomd 226 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥)) |
8 | 7 | pm5.74i 274 | . . 3 ⊢ ((𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
9 | 8 | albii 1821 | . 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
10 | 5, 9 | bitrdi 290 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 |
This theorem is referenced by: clel4 3578 intprg 4874 |
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