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Mirrors > Home > MPE Home > Th. List > fvnobday | Structured version Visualization version GIF version |
Description: The value of a surreal at its birthday is ∅. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) |
Ref | Expression |
---|---|
fvnobday | ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayval 27708 | . . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴) | |
2 | nodmord 27713 | . . . 4 ⊢ (𝐴 ∈ No → Ord dom 𝐴) | |
3 | ordirr 6404 | . . . 4 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴) |
5 | 1, 4 | eqneltrd 2859 | . 2 ⊢ (𝐴 ∈ No → ¬ ( bday ‘𝐴) ∈ dom 𝐴) |
6 | ndmfv 6942 | . 2 ⊢ (¬ ( bday ‘𝐴) ∈ dom 𝐴 → (𝐴‘( bday ‘𝐴)) = ∅) | |
7 | 5, 6 | syl 17 | 1 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ∅c0 4339 dom cdm 5689 Ord word 6385 ‘cfv 6563 No csur 27699 bday cbday 27701 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-no 27702 df-bday 27704 |
This theorem is referenced by: nodense 27752 |
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