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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 8518 | . 2 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5611 | 1 ⊢ (𝐴 ≺ ω → ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 ωcom 7582 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dom 8513 df-sdom 8514 |
This theorem is referenced by: unfi2 8789 unifi2 8816 |
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