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Mirrors > Home > MPE Home > Th. List > elex2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of elex2 2816 as of 30-Nov-2024. (Contributed by Alan Sare, 25-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
elex2OLD | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1a 2834 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | alrimiv 1925 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | elisset 2821 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | exim 1831 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵)) | |
5 | 2, 3, 4 | sylc 65 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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: (None) |
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