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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 8541 | . . . . . . . 8 ⊢ (rank‘𝑦) ∈ On | |
2 | 1 | onordi 5749 | . . . . . . 7 ⊢ Ord (rank‘𝑦) |
3 | eloni 5650 | . . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | ordsucsssuc 6915 | . . . . . . 7 ⊢ ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥)) | |
5 | 2, 3, 4 | sylancr 694 | . . . . . 6 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥)) |
6 | 1 | onsuci 6930 | . . . . . . 7 ⊢ suc (rank‘𝑦) ∈ On |
7 | suceloni 6905 | . . . . . . 7 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | r1ord3 8528 | . . . . . . 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 694 | . . . . . 6 ⊢ (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥))) |
10 | 5, 9 | sylbid 229 | . . . . 5 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥))) |
11 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
12 | 11 | rankid 8579 | . . . . 5 ⊢ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) |
13 | ssel 3562 | . . . . 5 ⊢ ((𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥) → (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑥))) | |
14 | 10, 12, 13 | syl6mpi 65 | . . . 4 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → 𝑦 ∈ (𝑅1‘suc 𝑥))) |
15 | 14 | ralimdv 2946 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))) |
16 | dfss3 3558 | . . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)) | |
17 | fvex 6113 | . . . . 5 ⊢ (𝑅1‘suc 𝑥) ∈ V | |
18 | 17 | ssex 4730 | . . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V) |
19 | 16, 18 | sylbir 224 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V) |
20 | 15, 19 | syl6 34 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)) |
21 | 20 | rexlimiv 3009 | 1 ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 Ord word 5639 Oncon0 5640 suc csuc 5642 ‘cfv 5804 𝑅1cr1 8508 rankcrnk 8509 |
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 ax-reg 8380 ax-inf2 8421 |
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-int 4411 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: unbndrank 8588 scottex 8631 |
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