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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 3732 | . 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω) | |
2 | bnj923.1 | . 2 ⊢ 𝐷 = (ω ∖ {∅}) | |
3 | 1, 2 | eleq2s 2719 | 1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∅c0 3915 {csn 4177 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 |
This theorem is referenced by: bnj1098 30854 bnj544 30964 bnj546 30966 bnj594 30982 bnj580 30983 bnj966 31014 bnj967 31015 bnj970 31017 bnj1001 31028 bnj1053 31044 bnj1071 31045 bnj1118 31052 bnj1128 31058 bnj1145 31061 |
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