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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 5684 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | rankxpl.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
3 | rankxpl.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | unex 7471 | . . . . . 6 ⊢ (𝐴 ∪ 𝐵) ∈ V |
5 | 4 | pwex 5283 | . . . . 5 ⊢ 𝒫 (𝐴 ∪ 𝐵) ∈ V |
6 | 5 | pwex 5283 | . . . 4 ⊢ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V |
7 | 6 | rankss 9280 | . . 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 9274 | . . 3 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘𝒫 (𝐴 ∪ 𝐵)) |
10 | 4 | rankpw 9274 | . . . 4 ⊢ (rank‘𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘(𝐴 ∪ 𝐵)) |
11 | suceq 6258 | . . . 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 2846 | . 2 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
14 | 8, 13 | sseqtri 4005 | 1 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 suc csuc 6195 ‘cfv 6357 rankcrnk 9194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-r1 9195 df-rank 9196 |
This theorem is referenced by: rankfu 9308 rankmapu 9309 rankxplim3 9312 |
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