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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 8661 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
2 | ssrab2 3650 | . . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On | |
3 | fvex 6113 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
4 | 1, 3 | syl6eqelr 2697 | . . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) |
5 | intex 4747 | . . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) | |
6 | 4, 5 | sylibr 223 | . . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) |
7 | onint 6887 | . . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
8 | 2, 6, 7 | sylancr 694 | . . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
9 | 1, 8 | eqeltrd 2688 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
10 | breq1 4586 | . . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴)) | |
11 | 10 | elrab 3331 | . . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴)) |
12 | 11 | simprbi 479 | . 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 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∩ cint 4410 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 ≈ cen 7838 cardccrd 8644 |
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-sep 4709 ax-nul 4717 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-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 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-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-en 7842 df-card 8648 |
This theorem is referenced by: isnum3 8663 oncardid 8665 cardidm 8668 ficardom 8670 ficardid 8671 cardnn 8672 cardnueq0 8673 carden2a 8675 carden2b 8676 carddomi2 8679 sdomsdomcardi 8680 cardsdomelir 8682 cardsdomel 8683 infxpidm2 8723 dfac8b 8737 numdom 8744 alephnbtwn2 8778 alephsucdom 8785 infenaleph 8797 dfac12r 8851 cardacda 8903 pwsdompw 8909 cff1 8963 cfflb 8964 cflim2 8968 cfss 8970 cfslb 8971 domtriomlem 9147 cardid 9248 cardidg 9249 carden 9252 sdomsdomcard 9261 hargch 9374 gch2 9376 hashkf 12981 |
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