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Theorem pm54.43lem 9997
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 9965), 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 9998. (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 9964 . . . 4 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
2 1onn 8641 . . . . 5 1o ∈ Ο‰
3 cardnn 9960 . . . . 5 (1o ∈ Ο‰ β†’ (cardβ€˜1o) = 1o)
42, 3ax-mp 5 . . . 4 (cardβ€˜1o) = 1o
51, 4eqtrdi 2782 . . 3 (𝐴 β‰ˆ 1o β†’ (cardβ€˜π΄) = 1o)
64eqeq2i 2739 . . . . 5 ((cardβ€˜π΄) = (cardβ€˜1o) ↔ (cardβ€˜π΄) = 1o)
76biimpri 227 . . . 4 ((cardβ€˜π΄) = 1o β†’ (cardβ€˜π΄) = (cardβ€˜1o))
8 1n0 8489 . . . . . . . 8 1o β‰  βˆ…
98neii 2936 . . . . . . 7 Β¬ 1o = βˆ…
10 eqeq1 2730 . . . . . . 7 ((cardβ€˜π΄) = 1o β†’ ((cardβ€˜π΄) = βˆ… ↔ 1o = βˆ…))
119, 10mtbiri 327 . . . . . 6 ((cardβ€˜π΄) = 1o β†’ Β¬ (cardβ€˜π΄) = βˆ…)
12 ndmfv 6920 . . . . . 6 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
1311, 12nsyl2 141 . . . . 5 ((cardβ€˜π΄) = 1o β†’ 𝐴 ∈ dom card)
14 1on 8479 . . . . . 6 1o ∈ On
15 onenon 9946 . . . . . 6 (1o ∈ On β†’ 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 9984 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
1813, 16, 17sylancl 585 . . . 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 6894 . . 3 (π‘₯ = 𝐴 β†’ ((cardβ€˜π‘₯) = 1o ↔ (cardβ€˜π΄) = 1o))
2213, 21elab3 3671 . 2 (𝐴 ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = 1o} ↔ (cardβ€˜π΄) = 1o)
2320, 22bitr4i 278 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 2703  βˆ…c0 4317   class class class wbr 5141  dom cdm 5669  Oncon0 6358  β€˜cfv 6537  Ο‰com 7852  1oc1o 8460   β‰ˆ cen 8938  cardccrd 9932
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936
This theorem is referenced by: (None)
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