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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj216 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bnj216.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| bnj216 | ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj216.1 | . . 3 ⊢ 𝐵 ∈ V | |
| 2 | 1 | sucid 6465 | . 2 ⊢ 𝐵 ∈ suc 𝐵 | 
| 3 | eleq2 2829 | . 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵)) | |
| 4 | 2, 3 | mpbiri 258 | 1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-suc 6389 | 
| This theorem is referenced by: bnj219 34748 bnj1098 34798 bnj556 34915 bnj557 34916 bnj594 34927 bnj944 34953 bnj966 34959 bnj969 34961 bnj1145 35008 | 
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