Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bndrank | Structured version Visualization version GIF version |
Description: Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
Ref | Expression |
---|---|
bndrank | ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankon 9212 | . . . . . . . 8 ⊢ (rank‘𝑦) ∈ On | |
2 | 1 | onordi 6288 | . . . . . . 7 ⊢ Ord (rank‘𝑦) |
3 | eloni 6194 | . . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | ordsucsssuc 7527 | . . . . . . 7 ⊢ ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥)) | |
5 | 2, 3, 4 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥)) |
6 | 1 | onsuci 7542 | . . . . . . 7 ⊢ suc (rank‘𝑦) ∈ On |
7 | suceloni 7517 | . . . . . . 7 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | r1ord3 9199 | . . . . . . 7 ⊢ ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥))) | |
9 | 6, 7, 8 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥))) |
10 | 5, 9 | sylbid 241 | . . . . 5 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥))) |
11 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
12 | 11 | rankid 9250 | . . . . 5 ⊢ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) |
13 | ssel 3958 | . . . . 5 ⊢ ((𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥) → (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑥))) | |
14 | 10, 12, 13 | syl6mpi 67 | . . . 4 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → 𝑦 ∈ (𝑅1‘suc 𝑥))) |
15 | 14 | ralimdv 3175 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))) |
16 | dfss3 3953 | . . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)) | |
17 | fvex 6676 | . . . . 5 ⊢ (𝑅1‘suc 𝑥) ∈ V | |
18 | 17 | ssex 5216 | . . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V) |
19 | 16, 18 | sylbir 236 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V) |
20 | 15, 19 | syl6 35 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)) |
21 | 20 | rexlimiv 3277 | 1 ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 Ord word 6183 Oncon0 6184 suc csuc 6186 ‘cfv 6348 𝑅1cr1 9179 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: unbndrank 9259 scottex 9302 |
Copyright terms: Public domain | W3C validator |