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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj923 | 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 | 
|---|---|
| bnj923.1 | ⊢ 𝐷 = (ω ∖ {∅}) | 
| Ref | Expression | 
|---|---|
| bnj923 | ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldifi 4130 | . 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω) | |
| 2 | bnj923.1 | . 2 ⊢ 𝐷 = (ω ∖ {∅}) | |
| 3 | 1, 2 | eleq2s 2858 | 1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ∅c0 4332 {csn 4625 ωcom 7888 | 
| 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-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 | 
| This theorem is referenced by: bnj1098 34798 bnj544 34909 bnj546 34911 bnj594 34927 bnj580 34928 bnj966 34959 bnj967 34960 bnj970 34962 bnj1001 34974 bnj1053 34991 bnj1071 34992 bnj1118 34999 bnj1128 35005 bnj1145 35008 | 
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