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Mirrors > Home > MPE Home > Th. List > oldbday | Structured version Visualization version GIF version |
Description: A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
oldbday | ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oldbdayim 27942 | . 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴) | |
2 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝐴 ∈ On) | |
3 | onelon 6411 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On) | |
4 | madebday 27953 | . . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( M ‘𝑏) ↔ ( bday ‘𝑦) ⊆ 𝑏)) | |
5 | 4 | biimprd 248 | . . . . . . 7 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) |
6 | 3, 5 | sylan 580 | . . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) |
7 | 6 | anasss 466 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑦 ∈ No )) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) |
8 | 7 | ralrimivva 3200 | . . . 4 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏))) |
10 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝑋 ∈ No ) | |
11 | madebdaylemold 27951 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴))) | |
12 | 2, 9, 10, 11 | syl3anc 1370 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴))) |
13 | 1, 12 | impbid2 226 | 1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 Oncon0 6386 ‘cfv 6563 No csur 27699 bday cbday 27701 M cmade 27896 O cold 27897 |
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-rep 5285 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-3or 1087 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-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 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-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 df-scut 27843 df-made 27901 df-old 27902 df-left 27904 df-right 27905 |
This theorem is referenced by: newbday 27955 0elold 27962 cofcutr 27973 lrrecval2 27988 addsproplem2 28018 addsproplem4 28020 addsproplem5 28021 addsproplem6 28022 negsproplem4 28078 negsproplem5 28079 negsproplem6 28080 mulsproplem12 28168 mulsproplem13 28169 mulsproplem14 28170 sltonold 28298 n0ssold 28370 |
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