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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 6273 | . 2 ⊢ 𝐵 ∈ suc 𝐵 |
3 | eleq2 2904 | . 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵)) | |
4 | 2, 3 | mpbiri 260 | 1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 suc csuc 6196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-un 3944 df-sn 4571 df-suc 6200 |
This theorem is referenced by: bnj219 32007 bnj1098 32059 bnj556 32176 bnj557 32177 bnj594 32188 bnj944 32214 bnj966 32220 bnj969 32222 bnj1145 32269 |
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