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Mirrors > Home > MPE Home > Th. List > rankxplim2 | Structured version Visualization version GIF version |
Description: If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.) |
Ref | Expression |
---|---|
rankxplim.1 | ⊢ 𝐴 ∈ V |
rankxplim.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankxplim2 | ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ellim 6221 | . . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵))) | |
2 | n0i 4249 | . . . 4 ⊢ (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅) |
4 | df-ne 2988 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅) | |
5 | rankxplim.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
6 | rankxplim.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | xpex 7456 | . . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V |
8 | 7 | rankeq0 9274 | . . . . 5 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅) |
9 | 8 | notbii 323 | . . . 4 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅) |
10 | 4, 9 | bitr2i 279 | . . 3 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅) |
11 | 3, 10 | sylib 221 | . 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅) |
12 | limuni2 6220 | . . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵))) | |
13 | limuni2 6220 | . . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵))) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵))) |
15 | rankuni 9276 | . . . . . 6 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵)) | |
16 | rankuni 9276 | . . . . . . 7 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)) | |
17 | 16 | unieqi 4813 | . . . . . 6 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)) |
18 | 15, 17 | eqtr2i 2822 | . . . . 5 ⊢ ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘∪ ∪ (𝐴 × 𝐵)) |
19 | unixp 6101 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | |
20 | 19 | fveq2d 6649 | . . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) |
21 | 18, 20 | syl5eq 2845 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) |
22 | limeq 6171 | . . . 4 ⊢ (∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)) → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵)))) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵)))) |
24 | 14, 23 | syl5ib 247 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))) |
25 | 11, 24 | mpcom 38 | 1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∪ cun 3879 ∅c0 4243 ∪ cuni 4800 × cxp 5517 Lim wlim 6160 ‘cfv 6324 rankcrnk 9176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 |
This theorem is referenced by: rankxpsuc 9295 |
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