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Theorem pm54.43lem 10021
Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9989), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o}. Here we show that this is equivalent to 𝐴 β‰ˆ 1o so that we can use the latter more convenient notation in pm54.43 10022. (Contributed by NM, 4-Nov-2013.)
Assertion
Ref Expression
pm54.43lem (𝐴 β‰ˆ 1o ↔ 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem pm54.43lem
StepHypRef Expression
1 carden2b 9988 . . . 4 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
2 1onn 8657 . . . . 5 1o ∈ Ο‰
3 cardnn 9984 . . . . 5 (1o ∈ Ο‰ β†’ (cardβ€˜1o) = 1o)
42, 3ax-mp 5 . . . 4 (cardβ€˜1o) = 1o
51, 4eqtrdi 2781 . . 3 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = 1o)
64eqeq2i 2738 . . . . 5 ((cardβ€˜π΄) = (cardβ€˜1o) ↔ (cardβ€˜π΄) = 1o)
76biimpri 227 . . . 4 ((cardβ€˜π΄) = 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
8 1n0 8505 . . . . . . . 8 1o β‰  βˆ…
98neii 2932 . . . . . . 7 Β¬ 1o = βˆ…
10 eqeq1 2729 . . . . . . 7 ((cardβ€˜π΄) = 1o β†’ ((cardβ€˜π΄) = βˆ… ↔ 1o = βˆ…))
119, 10mtbiri 326 . . . . . 6 ((cardβ€˜π΄) = 1o β†’ Β¬ (cardβ€˜π΄) = βˆ…)
12 ndmfv 6926 . . . . . 6 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
1311, 12nsyl2 141 . . . . 5 ((cardβ€˜π΄) = 1o β†’ 𝐴 ∈ dom card)
14 1on 8495 . . . . . 6 1o ∈ On
15 onenon 9970 . . . . . 6 (1o ∈ On β†’ 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 10008 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
1813, 16, 17sylancl 584 . . . 4 ((cardβ€˜π΄) = 1o β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
197, 18mpbid 231 . . 3 ((cardβ€˜π΄) = 1o β†’ 𝐴 β‰ˆ 1o)
205, 19impbii 208 . 2 (𝐴 β‰ˆ 1o ↔ (cardβ€˜π΄) = 1o)
21 fveqeq2 6900 . . 3 (π‘₯ = 𝐴 β†’ ((cardβ€˜π‘₯) = 1o ↔ (cardβ€˜π΄) = 1o))
2213, 21elab3 3668 . 2 (𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o} ↔ (cardβ€˜π΄) = 1o)
2320, 22bitr4i 277 1 (𝐴 β‰ˆ 1o ↔ 𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o})
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆ…c0 4318   class class class wbr 5143  dom cdm 5672  Oncon0 6364  β€˜cfv 6542  Ο‰com 7867  1oc1o 8476   β‰ˆ cen 8957  cardccrd 9956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7868  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960
This theorem is referenced by: (None)
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