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Mirrors > Home > MPE Home > Th. List > oldssmade | Structured version Visualization version GIF version |
Description: The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.) |
Ref | Expression |
---|---|
oldssmade | ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elold 27287 | . . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑥 ∈ ( M ‘𝑏))) | |
2 | onelss 6395 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑏 ∈ 𝐴 → 𝑏 ⊆ 𝐴)) | |
3 | 2 | imp 407 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ⊆ 𝐴) |
4 | madess 27294 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ⊆ 𝐴) → ( M ‘𝑏) ⊆ ( M ‘𝐴)) | |
5 | 3, 4 | syldan 591 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → ( M ‘𝑏) ⊆ ( M ‘𝐴)) |
6 | 5 | sseld 3977 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → (𝑥 ∈ ( M ‘𝑏) → 𝑥 ∈ ( M ‘𝐴))) |
7 | 6 | rexlimdva 3154 | . . . 4 ⊢ (𝐴 ∈ On → (∃𝑏 ∈ 𝐴 𝑥 ∈ ( M ‘𝑏) → 𝑥 ∈ ( M ‘𝐴))) |
8 | 1, 7 | sylbid 239 | . . 3 ⊢ (𝐴 ∈ On → (𝑥 ∈ ( O ‘𝐴) → 𝑥 ∈ ( M ‘𝐴))) |
9 | 8 | ssrdv 3984 | . 2 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴)) |
10 | oldf 27275 | . . . . . 6 ⊢ O :On⟶𝒫 No | |
11 | 10 | fdmi 6716 | . . . . 5 ⊢ dom O = On |
12 | 11 | eleq2i 2824 | . . . 4 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On) |
13 | ndmfv 6913 | . . . 4 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 330 | . . 3 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅) |
15 | 0ss 4392 | . . 3 ⊢ ∅ ⊆ ( M ‘𝐴) | |
16 | 14, 15 | eqsstrdi 4032 | . 2 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴)) |
17 | 9, 16 | pm2.61i 182 | 1 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4596 dom cdm 5669 Oncon0 6353 ‘cfv 6532 No csur 27070 M cmade 27260 O cold 27261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-1o 8448 df-2o 8449 df-no 27073 df-slt 27074 df-bday 27075 df-sslt 27209 df-scut 27211 df-made 27265 df-old 27266 |
This theorem is referenced by: madeun 27301 madeoldsuc 27302 oldlim 27304 |
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