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Mirrors > Home > MPE Home > Th. List > ficardun2 | Structured version Visualization version GIF version |
Description: The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.) |
Ref | Expression |
---|---|
ficardun2 | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncdadom 8876 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵)) | |
2 | finnum 8657 | . . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
3 | finnum 8657 | . . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
4 | cardacda 8903 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) | |
5 | 2, 3, 4 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
6 | domentr 7901 | . . . 4 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵))) | |
7 | 1, 5, 6 | syl2anc 691 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵))) |
8 | unfi 8112 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
9 | finnum 8657 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∪ 𝐵) ∈ dom card) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ dom card) |
11 | ficardom 8670 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
12 | ficardom 8670 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω) | |
13 | nnacl 7578 | . . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω) | |
14 | 11, 12, 13 | syl2an 493 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω) |
15 | nnon 6963 | . . . . 5 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On) | |
16 | onenon 8658 | . . . . 5 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card) | |
17 | 14, 15, 16 | 3syl 18 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card) |
18 | carddom2 8686 | . . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))) | |
19 | 10, 17, 18 | syl2anc 691 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))) |
20 | 7, 19 | mpbird 246 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵)))) |
21 | cardnn 8672 | . . 3 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) | |
22 | 14, 21 | syl 17 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵))) |
23 | 20, 22 | sseqtrd 3604 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 ωcom 6957 +𝑜 coa 7444 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 cardccrd 8644 +𝑐 ccda 8872 |
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 |
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-rmo 2904 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-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 |
This theorem is referenced by: (None) |
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