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 3203 | . . 3 ⊢ 𝑚 ∈ V | |
2 | 1 | bnj216 30800 | . 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛) |
3 | epel 5032 | . 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛) | |
4 | 2, 3 | sylibr 224 | 1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 class class class wbr 4653 E cep 5028 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-suc 5729 |
This theorem is referenced by: bnj605 30977 bnj594 30982 bnj607 30986 bnj1110 31050 |
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