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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 6127 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | |
2 | 1 | fveq2d 6668 | . 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 7470 | . . . . 5 ⊢ (𝐴 × 𝐵) ∈ V |
6 | 5 | uniex 7461 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ∈ V |
7 | 6 | rankuniss 9289 | . . 3 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘∪ (𝐴 × 𝐵)) |
8 | 5 | rankuniss 9289 | . . 3 ⊢ (rank‘∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
9 | 7, 8 | sstri 3975 | . 2 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
10 | 2, 9 | eqsstrrdi 4021 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 × cxp 5547 ‘cfv 6349 rankcrnk 9186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-reg 9050 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-r1 9187 df-rank 9188 |
This theorem is referenced by: rankxplim 9302 |
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