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Mirrors > Home > MPE Home > Th. List > cardacda | Structured version Visualization version GIF version |
Description: The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
cardacda | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 8770 | . . . 4 ⊢ (card‘𝐴) ∈ On | |
2 | cardon 8770 | . . . 4 ⊢ (card‘𝐵) ∈ On | |
3 | onacda 9019 | . . . 4 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵))) | |
4 | 1, 2, 3 | mp2an 708 | . . 3 ⊢ ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵)) |
5 | cardid2 8779 | . . . 4 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
6 | cardid2 8779 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | cdaen 8995 | . . . 4 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) | |
8 | 5, 6, 7 | syl2an 494 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) |
9 | entr 8008 | . . 3 ⊢ ((((card‘𝐴) +𝑜 (card‘𝐵)) ≈ ((card‘𝐴) +𝑐 (card‘𝐵)) ∧ ((card‘𝐴) +𝑐 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) | |
10 | 4, 8, 9 | sylancr 695 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵)) |
11 | 10 | ensymd 8007 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 Oncon0 5723 ‘cfv 5888 (class class class)co 6650 +𝑜 coa 7557 ≈ cen 7952 cardccrd 8761 +𝑐 ccda 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-card 8765 df-cda 8990 |
This theorem is referenced by: cdanum 9021 ficardun 9024 ficardun2 9025 pwsdompw 9026 |
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