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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 9609 | . . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankr1b.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
3 | 2 | rankuni2 9601 | . . . . 5 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) |
4 | 1, 3 | eqtr3i 2768 | . . . 4 ⊢ ∪ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) |
5 | 4 | sseq1i 3949 | . . 3 ⊢ (∪ (rank‘𝐴) ⊆ 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵) |
6 | iunss 4975 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵) | |
7 | 5, 6 | bitr2i 275 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ ∪ (rank‘𝐴) ⊆ 𝐵) |
8 | rankon 9541 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
9 | 8 | onssi 7675 | . . 3 ⊢ (rank‘𝐴) ⊆ On |
10 | eloni 6270 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
11 | ordunisssuc 6362 | . . 3 ⊢ (((rank‘𝐴) ⊆ On ∧ Ord 𝐵) → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) | |
12 | 9, 10, 11 | sylancr 587 | . 2 ⊢ (𝐵 ∈ On → (∪ (rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) |
13 | 7, 12 | bitrid 282 | 1 ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∀wral 3064 Vcvv 3430 ⊆ wss 3887 ∪ cuni 4840 ∪ ciun 4925 Ord word 6259 Oncon0 6260 suc csuc 6262 ‘cfv 6427 rankcrnk 9509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-reg 9339 ax-inf2 9387 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-r1 9510 df-rank 9511 |
This theorem is referenced by: (None) |
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