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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 6302 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | |
| 2 | 1 | fveq2d 6910 | . 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 7773 | . . . . 5 ⊢ (𝐴 × 𝐵) ∈ V |
| 6 | 5 | uniex 7761 | . . . 4 ⊢ ∪ (𝐴 × 𝐵) ∈ V |
| 7 | 6 | rankuniss 9906 | . . 3 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘∪ (𝐴 × 𝐵)) |
| 8 | 5 | rankuniss 9906 | . . 3 ⊢ (rank‘∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
| 9 | 7, 8 | sstri 3993 | . 2 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) |
| 10 | 2, 9 | eqsstrrdi 4029 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 × cxp 5683 ‘cfv 6561 rankcrnk 9803 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 |
| This theorem is referenced by: rankxplim 9919 |
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