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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 8250 | . 2 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5409 | 1 ⊢ (𝐴 ≺ ω → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3398 class class class wbr 4888 ωcom 7345 ≺ csdm 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-xp 5363 df-rel 5364 df-dom 8245 df-sdom 8246 |
This theorem is referenced by: unfi2 8519 unifi2 8546 |
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