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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 8510 | . 2 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5604 | 1 ⊢ (𝐴 ≺ ω → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 ωcom 7574 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-dom 8505 df-sdom 8506 |
This theorem is referenced by: unfi2 8781 unifi2 8808 |
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