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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 3694 | . 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω) | |
2 | bnj923.1 | . 2 ⊢ 𝐷 = (ω ∖ {∅}) | |
3 | 1, 2 | eleq2s 2706 | 1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 {csn 4125 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 |
This theorem is referenced by: bnj1098 30108 bnj544 30218 bnj546 30220 bnj594 30236 bnj580 30237 bnj966 30268 bnj967 30269 bnj970 30271 bnj1001 30282 bnj1053 30298 bnj1071 30299 bnj1118 30306 bnj1128 30312 bnj1145 30315 |
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