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Mirrors > Home > MPE Home > Th. List > rankmapu | Structured version Visualization version GIF version |
Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.) |
Ref | Expression |
---|---|
rankxpl.1 | ⊢ 𝐴 ∈ V |
rankxpl.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankmapu | ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsspw 7779 | . . 3 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
2 | rankxpl.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | rankxpl.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | xpex 6860 | . . . . 5 ⊢ (𝐵 × 𝐴) ∈ V |
5 | 4 | pwex 4774 | . . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V |
6 | 5 | rankss 8595 | . . 3 ⊢ ((𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) → (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴))) |
7 | 1, 6 | ax-mp 5 | . 2 ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴)) |
8 | 4 | rankpw 8589 | . . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴)) |
9 | 2, 3 | rankxpu 8622 | . . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴)) |
10 | uncom 3719 | . . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
11 | 10 | fveq2i 6106 | . . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) |
12 | suceq 5707 | . . . . . . 7 ⊢ ((rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) → suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)) |
14 | suceq 5707 | . . . . . 6 ⊢ (suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)) → suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵))) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵)) |
16 | 9, 15 | sseqtri 3600 | . . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
17 | rankon 8541 | . . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On | |
18 | 17 | onordi 5749 | . . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴)) |
19 | rankon 8541 | . . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On | |
20 | 19 | onsuci 6930 | . . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
21 | 20 | onsuci 6930 | . . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
22 | 21 | onordi 5749 | . . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵)) |
23 | ordsucsssuc 6915 | . . . . 5 ⊢ ((Ord (rank‘(𝐵 × 𝐴)) ∧ Ord suc suc (rank‘(𝐴 ∪ 𝐵))) → ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)))) | |
24 | 18, 22, 23 | mp2an 704 | . . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))) |
25 | 16, 24 | mpbi 219 | . . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
26 | 8, 25 | eqsstri 3598 | . 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
27 | 7, 26 | sstri 3577 | 1 ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 × cxp 5036 Ord word 5639 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 rankcrnk 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-map 7746 df-pm 7747 df-r1 8510 df-rank 8511 |
This theorem is referenced by: (None) |
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