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Mirrors > Home > MPE Home > Th. List > cardid | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cardval.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
cardid | ⊢ (card‘𝐴) ≈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardval.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | numth3 9895 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
3 | cardid2 9385 | . 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 2113 Vcvv 3497 class class class wbr 5069 dom cdm 5558 ‘cfv 6358 ≈ cen 8509 cardccrd 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-ac2 9888 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-wrecs 7950 df-recs 8011 df-en 8513 df-card 9371 df-ac 9545 |
This theorem is referenced by: unsnen 9978 alephval2 9997 cfpwsdom 10009 inar1 10200 gruina 10243 |
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