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Mirrors > Home > MPE Home > Th. List > cardid2 | Structured version Visualization version GIF version |
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
Ref | Expression |
---|---|
cardid2 | ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardval3 9369 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
2 | ssrab2 4053 | . . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On | |
3 | fvex 6676 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
4 | 1, 3 | syl6eqelr 2919 | . . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) |
5 | intex 5231 | . . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) | |
6 | 4, 5 | sylibr 235 | . . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) |
7 | onint 7499 | . . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
8 | 2, 6, 7 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
9 | 1, 8 | eqeltrd 2910 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
10 | breq1 5060 | . . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴)) | |
11 | 10 | elrab 3677 | . . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴)) |
12 | 11 | simprbi 497 | . 2 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → (card‘𝐴) ≈ 𝐴) |
13 | 9, 12 | syl 17 | 1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 ∩ cint 4867 class class class wbr 5057 dom cdm 5548 Oncon0 6184 ‘cfv 6348 ≈ cen 8494 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-en 8498 df-card 9356 |
This theorem is referenced by: isnum3 9371 oncardid 9373 cardidm 9376 ficardom 9378 ficardid 9379 cardnn 9380 cardnueq0 9381 carden2a 9383 carden2b 9384 carddomi2 9387 sdomsdomcardi 9388 cardsdomelir 9390 cardsdomel 9391 infxpidm2 9431 dfac8b 9445 numdom 9452 alephnbtwn2 9486 alephsucdom 9493 infenaleph 9505 dfac12r 9560 cardadju 9608 pwsdompw 9614 cff1 9668 cfflb 9669 cflim2 9673 cfss 9675 cfslb 9676 domtriomlem 9852 cardid 9957 cardidg 9958 carden 9961 sdomsdomcard 9970 hargch 10083 gch2 10085 hashkf 13680 |
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