![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8236 | . . 3 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
2 | rankxpl.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | rankxpl.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | xpex 7287 | . . . . 5 ⊢ (𝐵 × 𝐴) ∈ V |
5 | 4 | pwex 5128 | . . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V |
6 | 5 | rankss 9066 | . . 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 9060 | . . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴)) |
9 | 2, 3 | rankxpu 9093 | . . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴)) |
10 | uncom 4012 | . . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
11 | 10 | fveq2i 6496 | . . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) |
12 | suceq 6088 | . . . . . . 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 6088 | . . . . . 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 3887 | . . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
17 | rankon 9012 | . . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On | |
18 | 17 | onordi 6127 | . . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴)) |
19 | rankon 9012 | . . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On | |
20 | 19 | onsuci 7363 | . . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
21 | 20 | onsuci 7363 | . . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
22 | 21 | onordi 6127 | . . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵)) |
23 | ordsucsssuc 7348 | . . . . 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 679 | . . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))) |
25 | 16, 24 | mpbi 222 | . . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
26 | 8, 25 | eqsstri 3885 | . 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
27 | 7, 26 | sstri 3861 | 1 ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ∪ cun 3821 ⊆ wss 3823 𝒫 cpw 4416 × cxp 5399 Ord word 6022 suc csuc 6025 ‘cfv 6182 (class class class)co 6970 ↑𝑚 cmap 8200 rankcrnk 8980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-reg 8845 ax-inf2 8892 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-map 8202 df-pm 8203 df-r1 8981 df-rank 8982 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |