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Mirrors > Home > MPE Home > Th. List > rankbnd2 | Structured version Visualization version GIF version |
Description: The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.) |
Ref | Expression |
---|---|
rankr1b.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rankbnd2 | ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankuni 9276 | . . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankr1b.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
3 | 2 | rankuni2 9268 | . . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) |
4 | 1, 3 | eqtr3i 2823 | . . . 4 ⊢ ∪ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) |
5 | 4 | sseq1i 3943 | . . 3 ⊢ (∪ (rank‘𝐴) ⊆ 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵) |
6 | iunss 4932 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵) | |
7 | 5, 6 | bitr2i 279 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∪ (rank‘𝐴) ⊆ 𝐵) |
8 | rankon 9208 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
9 | 8 | onssi 7532 | . . 3 ⊢ (rank‘𝐴) ⊆ On |
10 | eloni 6169 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
11 | ordunisssuc 6261 | . . 3 ⊢ (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) | |
12 | 9, 10, 11 | sylancr 590 | . 2 ⊢ (𝐵 ∈ On → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) |
13 | 7, 12 | syl5bb 286 | 1 ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 ∪ ciun 4881 Ord word 6158 Oncon0 6159 suc csuc 6161 ‘cfv 6324 rankcrnk 9176 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 |
This theorem is referenced by: (None) |
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