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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 5156 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | rankxpl.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
3 | rankxpl.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | unex 6854 | . . . . . 6 ⊢ (𝐴 ∪ 𝐵) ∈ V |
5 | 4 | pwex 4774 | . . . . 5 ⊢ 𝒫 (𝐴 ∪ 𝐵) ∈ V |
6 | 5 | pwex 4774 | . . . 4 ⊢ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V |
7 | 6 | rankss 8595 | . . 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 8589 | . . 3 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘𝒫 (𝐴 ∪ 𝐵)) |
10 | 4 | rankpw 8589 | . . . 4 ⊢ (rank‘𝒫 (𝐴 ∪ 𝐵)) = suc (rank‘(𝐴 ∪ 𝐵)) |
11 | suceq 5707 | . . . 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 2632 | . 2 ⊢ (rank‘𝒫 𝒫 (𝐴 ∪ 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
14 | 8, 13 | sseqtri 3600 | 1 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 × cxp 5036 suc csuc 5642 ‘cfv 5804 rankcrnk 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: rankfu 8623 rankmapu 8624 rankxplim3 8627 |
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