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Theorem cardid 10571

Description: Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Hypothesis**
Ref	Expression
cardval.1	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
cardid	⊢ (card‘𝐴) ≈ 𝐴

**Proof of Theorem cardid**
Step	Hyp	Ref	Expression
1		cardval.1	. 2 ⊢ 𝐴 ∈ V
2		numth3 10494	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card)
3		cardid2 9977	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4	1, 2, 3	mp2b 10	1 ⊢ (card‘𝐴) ≈ 𝐴

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2099 Vcvv 3471 class class class wbr 5148 dom cdm 5678 ‘cfv 6548 ≈ cen 8961 cardccrd 9959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-ac2 10487

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-en 8965 df-card 9963 df-ac 10140

This theorem is referenced by: unsnen 10577 alephval2 10596 cfpwsdom 10608 inar1 10799 gruina 10842

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