| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj219 | 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 |
|---|---|
| bnj219 | ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . 3 ⊢ 𝑚 ∈ V | |
| 2 | 1 | bnj216 34746 | . 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛) |
| 3 | epel 5587 | . 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 class class class wbr 5143 E cep 5583 suc csuc 6386 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-suc 6390 |
| This theorem is referenced by: bnj605 34921 bnj594 34926 bnj607 34930 bnj1110 34996 |
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