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Mirrors > Home > MPE Home > Th. List > Mathboxes > bdayelon | Structured version Visualization version GIF version |
Description: The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) |
Ref | Expression |
---|---|
bdayelon | ⊢ ( bday ‘𝐴) ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfun 31075 | . . 3 ⊢ Fun bday | |
2 | fvelrn 6260 | . . . 4 ⊢ ((Fun bday ∧ 𝐴 ∈ dom bday ) → ( bday ‘𝐴) ∈ ran bday ) | |
3 | bdayrn 31076 | . . . 4 ⊢ ran bday = On | |
4 | 2, 3 | syl6eleq 2698 | . . 3 ⊢ ((Fun bday ∧ 𝐴 ∈ dom bday ) → ( bday ‘𝐴) ∈ On) |
5 | 1, 4 | mpan 702 | . 2 ⊢ (𝐴 ∈ dom bday → ( bday ‘𝐴) ∈ On) |
6 | ndmfv 6128 | . . 3 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅) | |
7 | 0elon 5695 | . . 3 ⊢ ∅ ∈ On | |
8 | 6, 7 | syl6eqel 2696 | . 2 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) ∈ On) |
9 | 5, 8 | pm2.61i 175 | 1 ⊢ ( bday ‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∈ wcel 1977 ∅c0 3874 dom cdm 5038 ran crn 5039 Oncon0 5640 Fun wfun 5798 ‘cfv 5804 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: fvnobday 31081 nodenselem3 31082 nodenselem4 31083 nodenselem6 31085 nodense 31088 nocvxminlem 31089 nobndlem2 31092 nobndlem4 31094 nobndlem5 31095 nobndlem6 31096 nobndlem8 31098 |
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