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Mirrors > Home > MPE Home > Th. List > rankxpl | Structured version Visualization version GIF version |
Description: A lower 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 |
---|---|
rankxpl | ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unixp 6292 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | |
2 | 1 | fveq2d 6904 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) |
3 | rankxpl.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | rankxpl.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | xpex 7760 | . . . . 5 ⊢ (𝐴 × 𝐵) ∈ V |
6 | 5 | uniex 7751 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ∈ V |
7 | 6 | rankuniss 9905 | . . 3 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘∪ (𝐴 × 𝐵)) |
8 | 5 | rankuniss 9905 | . . 3 ⊢ (rank‘∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
9 | 7, 8 | sstri 3988 | . 2 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
10 | 2, 9 | eqsstrrdi 4034 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∪ cun 3944 ⊆ wss 3946 ∅c0 4324 ∪ cuni 4912 × cxp 5679 ‘cfv 6553 rankcrnk 9802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-reg 9631 ax-inf2 9680 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-r1 9803 df-rank 9804 |
This theorem is referenced by: rankxplim 9918 |
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