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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 3479 | . . 3 ⊢ 𝑚 ∈ V | |
2 | 1 | bnj216 33680 | . 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛) |
3 | epel 5581 | . 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 class class class wbr 5146 E cep 5577 suc csuc 6362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-br 5147 df-opab 5209 df-eprel 5578 df-suc 6366 |
This theorem is referenced by: bnj605 33855 bnj594 33860 bnj607 33864 bnj1110 33930 |
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