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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 | bdayelon 31079 | . . . 4 ⊢ ( bday ‘𝐴) ∈ On | |
2 | 1 | onirri 5751 | . . 3 ⊢ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐴) |
3 | bdayval 31045 | . . . 4 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴) | |
4 | 3 | eleq2d 2673 | . . 3 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) ∈ ( bday ‘𝐴) ↔ ( bday ‘𝐴) ∈ dom 𝐴)) |
5 | 2, 4 | mtbii 315 | . 2 ⊢ (𝐴 ∈ No → ¬ ( bday ‘𝐴) ∈ dom 𝐴) |
6 | ndmfv 6128 | . 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 1475 ∈ wcel 1977 ∅c0 3874 dom cdm 5038 ‘cfv 5804 No csur 31037 bday cbday 31039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-1o 7447 df-no 31040 df-bday 31042 |
This theorem is referenced by: nodenselem3 31082 nodense 31088 nobndlem2 31092 nobndlem4 31094 nobndlem5 31095 nobndlem6 31096 nobndlem8 31098 |
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