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Mirrors > Home > MPE Home > Th. List > clel4 | Structured version Visualization version GIF version |
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
clel4.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
clel4 | ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clel4.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | eleq2 2903 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ceqsalv 3534 | . 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥) ↔ 𝐴 ∈ 𝐵) |
4 | 3 | bicomi 226 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1781 df-nf 1785 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: intpr 4911 |
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