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Mirrors > Home > MPE Home > Th. List > rankmapu | Structured version Visualization version GIF version |
Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.) |
Ref | Expression |
---|---|
rankxpl.1 | ⊢ 𝐴 ∈ V |
rankxpl.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankmapu | ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsspw 7893 | . . 3 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
2 | rankxpl.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | rankxpl.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | xpex 6962 | . . . . 5 ⊢ (𝐵 × 𝐴) ∈ V |
5 | 4 | pwex 4848 | . . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V |
6 | 5 | rankss 8712 | . . 3 ⊢ ((𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) → (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴))) |
7 | 1, 6 | ax-mp 5 | . 2 ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴)) |
8 | 4 | rankpw 8706 | . . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴)) |
9 | 2, 3 | rankxpu 8739 | . . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴)) |
10 | uncom 3757 | . . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
11 | 10 | fveq2i 6194 | . . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) |
12 | suceq 5790 | . . . . . . 7 ⊢ ((rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) → suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)) |
14 | suceq 5790 | . . . . . 6 ⊢ (suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)) → suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵))) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
16 | 9, 15 | sseqtri 3637 | . . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
17 | rankon 8658 | . . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On | |
18 | 17 | onordi 5832 | . . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴)) |
19 | rankon 8658 | . . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On | |
20 | 19 | onsuci 7038 | . . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
21 | 20 | onsuci 7038 | . . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
22 | 21 | onordi 5832 | . . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵)) |
23 | ordsucsssuc 7023 | . . . . 5 ⊢ ((Ord (rank‘(𝐵 × 𝐴)) ∧ Ord suc suc (rank‘(𝐴 ∪ 𝐵))) → ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)))) | |
24 | 18, 22, 23 | mp2an 708 | . . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))) |
25 | 16, 24 | mpbi 220 | . . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
26 | 8, 25 | eqsstri 3635 | . 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
27 | 7, 26 | sstri 3612 | 1 ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 × cxp 5112 Ord word 5722 suc csuc 5725 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 rankcrnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-map 7859 df-pm 7860 df-r1 8627 df-rank 8628 |
This theorem is referenced by: (None) |
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