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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 5721 | . 2 ⊢ 𝐵 ∈ suc 𝐵 |
3 | eleq2 2677 | . 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵)) | |
4 | 2, 3 | mpbiri 247 | 1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 suc csuc 5642 |
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-un 3545 df-sn 4126 df-suc 5646 |
This theorem is referenced by: bnj219 30055 bnj1098 30108 bnj556 30224 bnj557 30225 bnj594 30236 bnj944 30262 bnj966 30268 bnj969 30270 bnj1145 30315 |
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