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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 3176 | . . 3 ⊢ 𝑚 ∈ V | |
2 | 1 | bnj216 30054 | . 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛) |
3 | epel 4952 | . 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛) | |
4 | 2, 3 | sylibr 223 | 1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 class class class wbr 4583 E cep 4947 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-suc 5646 |
This theorem is referenced by: bnj605 30231 bnj594 30236 bnj607 30240 bnj1110 30304 |
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