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Theorem pm54.43lem 10040
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 10008), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}. Here we show that this is equivalent to 𝐴 ≈ 1o so that we can use the latter more convenient notation in pm54.43 10041. (Contributed by NM, 4-Nov-2013.)
Assertion
Ref Expression
pm54.43lem (𝐴 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
Distinct variable group:   𝑥,𝐴

Proof of Theorem pm54.43lem
StepHypRef Expression
1 carden2b 10007 . . . 4 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2 1onn 8678 . . . . 5 1o ∈ ω
3 cardnn 10003 . . . . 5 (1o ∈ ω → (card‘1o) = 1o)
42, 3ax-mp 5 . . . 4 (card‘1o) = 1o
51, 4eqtrdi 2793 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = 1o)
64eqeq2i 2750 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
76biimpri 228 . . . 4 ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8 1n0 8526 . . . . . . . 8 1o ≠ ∅
98neii 2942 . . . . . . 7 ¬ 1o = ∅
10 eqeq1 2741 . . . . . . 7 ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
119, 10mtbiri 327 . . . . . 6 ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12 ndmfv 6941 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 141 . . . . 5 ((card‘𝐴) = 1o𝐴 ∈ dom card)
14 1on 8518 . . . . . 6 1o ∈ On
15 onenon 9989 . . . . . 6 (1o ∈ On → 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 10027 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
1813, 16, 17sylancl 586 . . . 4 ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
197, 18mpbid 232 . . 3 ((card‘𝐴) = 1o𝐴 ≈ 1o)
205, 19impbii 209 . 2 (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
21 fveqeq2 6915 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
2213, 21elab3 3686 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
2320, 22bitr4i 278 1 (𝐴 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  {cab 2714  c0 4333   class class class wbr 5143  dom cdm 5685  Oncon0 6384  cfv 6561  ωcom 7887  1oc1o 8499  cen 8982  cardccrd 9975
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979
This theorem is referenced by: (None)
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