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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 6449 | . . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵))) | |
2 | n0i 4346 | . . . 4 ⊢ (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅) |
4 | df-ne 2939 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅) | |
5 | rankxplim.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
6 | rankxplim.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | xpex 7772 | . . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V |
8 | 7 | rankeq0 9899 | . . . . 5 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅) |
9 | 8 | notbii 320 | . . . 4 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅) |
10 | 4, 9 | bitr2i 276 | . . 3 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅) |
11 | 3, 10 | sylib 218 | . 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅) |
12 | limuni2 6448 | . . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵))) | |
13 | limuni2 6448 | . . . 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 9901 | . . . . . 6 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵)) | |
16 | rankuni 9901 | . . . . . . 7 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)) | |
17 | 16 | unieqi 4924 | . . . . . 6 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)) |
18 | 15, 17 | eqtr2i 2764 | . . . . 5 ⊢ ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘∪ ∪ (𝐴 × 𝐵)) |
19 | unixp 6304 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | |
20 | 19 | fveq2d 6911 | . . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) |
21 | 18, 20 | eqtrid 2787 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) |
22 | limeq 6398 | . . . 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 | imbitrid 244 | . 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 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∪ cun 3961 ∅c0 4339 ∪ cuni 4912 × cxp 5687 Lim wlim 6387 ‘cfv 6563 rankcrnk 9801 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 df-rank 9803 |
This theorem is referenced by: rankxpsuc 9920 |
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