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Mirrors > Home > MPE Home > Th. List > elopabw | Structured version Visualization version GIF version |
Description: Membership in a class abstraction of ordered pairs. Weaker version of elopab 5482 with a sethood antecedent, avoiding ax-sep 5254, ax-nul 5261, and ax-pr 5382. Originally a subproof of elopab 5482. (Contributed by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
elopabw | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2741 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩)) | |
2 | 1 | anbi1d 630 | . . 3 ⊢ (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
3 | 2 | 2exbidv 1927 | . 2 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
4 | df-opab 5166 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} | |
5 | 3, 4 | elab2g 3630 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ⟨cop 4590 {copab 5165 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-opab 5166 |
This theorem is referenced by: elopab 5482 iunopab 5514 ssrel 5736 |
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