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Mirrors > Home > MPE Home > Th. List > fin2inf | Structured version Visualization version GIF version |
Description: This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless ω exists. (Contributed by NM, 13-Nov-2003.) |
Ref | Expression |
---|---|
fin2inf | ⊢ (𝐴 ≺ ω → ω ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8897 | . 2 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5694 | 1 ⊢ (𝐴 ≺ ω → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3448 class class class wbr 5110 ωcom 7807 ≺ csdm 8889 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: unfi2 9266 unifi2 9293 |
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