![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 27923 | . . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑥 ∈ ( M ‘𝑏))) | |
2 | onelss 6428 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑏 ∈ 𝐴 → 𝑏 ⊆ 𝐴)) | |
3 | 2 | imp 406 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ⊆ 𝐴) |
4 | madess 27930 | . . . . . . 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 3994 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → (𝑥 ∈ ( M ‘𝑏) → 𝑥 ∈ ( M ‘𝐴))) |
7 | 6 | rexlimdva 3153 | . . . 4 ⊢ (𝐴 ∈ On → (∃𝑏 ∈ 𝐴 𝑥 ∈ ( M ‘𝑏) → 𝑥 ∈ ( M ‘𝐴))) |
8 | 1, 7 | sylbid 240 | . . 3 ⊢ (𝐴 ∈ On → (𝑥 ∈ ( O ‘𝐴) → 𝑥 ∈ ( M ‘𝐴))) |
9 | 8 | ssrdv 4001 | . 2 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴)) |
10 | oldf 27911 | . . . . . 6 ⊢ O :On⟶𝒫 No | |
11 | 10 | fdmi 6748 | . . . . 5 ⊢ dom O = On |
12 | 11 | eleq2i 2831 | . . . 4 ⊢ (𝐴 ∈ dom O ↔ 𝐴 ∈ On) |
13 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝐴 ∈ dom O → ( O ‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 331 | . . 3 ⊢ (¬ 𝐴 ∈ On → ( O ‘𝐴) = ∅) |
15 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ ( M ‘𝐴) | |
16 | 14, 15 | eqsstrdi 4050 | . 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 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 dom cdm 5689 Oncon0 6386 ‘cfv 6563 No csur 27699 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 |
This theorem is referenced by: madeun 27937 madeoldsuc 27938 oldlim 27940 |
Copyright terms: Public domain | W3C validator |