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Mirrors > Home > MPE Home > Th. List > rankxpu | Structured version Visualization version GIF version |
Description: An upper bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
rankxpl.1 | ⊢ 𝐴 ∈ V |
rankxpl.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankxpu | ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsspw 5718 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | rankxpl.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
3 | rankxpl.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | unex 7591 | . . . . . 6 ⊢ (𝐴 ∪ 𝐵) ∈ V |
5 | 4 | pwex 5307 | . . . . 5 ⊢ 𝒫 (𝐴 ∪ 𝐵) ∈ V |
6 | 5 | pwex 5307 | . . . 4 ⊢ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V |
7 | 6 | rankss 9618 | . . 3 ⊢ ((𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵))) |
8 | 1, 7 | ax-mp 5 | . 2 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) |
9 | 5 | rankpw 9612 | . . 3 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘𝒫 (𝐴 ∪ 𝐵)) |
10 | 4 | rankpw 9612 | . . . 4 ⊢ (rank‘𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘(𝐴 ∪ 𝐵)) |
11 | suceq 6330 | . . . 4 ⊢ ((rank‘𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘(𝐴 ∪ 𝐵)) → suc (rank‘𝒫 (𝐴 ∪ 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ suc (rank‘𝒫 (𝐴 ∪ 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
13 | 9, 12 | eqtri 2768 | . 2 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
14 | 8, 13 | sseqtri 3962 | 1 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∪ cun 3890 ⊆ wss 3892 𝒫 cpw 4539 × cxp 5588 suc csuc 6267 ‘cfv 6432 rankcrnk 9532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 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 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7275 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-r1 9533 df-rank 9534 |
This theorem is referenced by: rankfu 9646 rankmapu 9647 rankxplim3 9650 |
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