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Theorem pm54.43lem 9462
 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 9430), 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 9463. (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 9429 . . . 4 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2 1onn 8275 . . . . 5 1o ∈ ω
3 cardnn 9425 . . . . 5 (1o ∈ ω → (card‘1o) = 1o)
42, 3ax-mp 5 . . . 4 (card‘1o) = 1o
51, 4eqtrdi 2809 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = 1o)
64eqeq2i 2771 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
76biimpri 231 . . . 4 ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8 1n0 8129 . . . . . . . 8 1o ≠ ∅
98neii 2953 . . . . . . 7 ¬ 1o = ∅
10 eqeq1 2762 . . . . . . 7 ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
119, 10mtbiri 330 . . . . . 6 ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12 ndmfv 6688 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 143 . . . . 5 ((card‘𝐴) = 1o𝐴 ∈ dom card)
14 1on 8119 . . . . . 6 1o ∈ On
15 onenon 9411 . . . . . 6 (1o ∈ On → 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 9449 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
1813, 16, 17sylancl 589 . . . 4 ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
197, 18mpbid 235 . . 3 ((card‘𝐴) = 1o𝐴 ≈ 1o)
205, 19impbii 212 . 2 (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
21 fveqeq2 6667 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
2213, 21elab3 3595 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
2320, 22bitr4i 281 1 (𝐴 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2735  ∅c0 4225   class class class wbr 5032  dom cdm 5524  Oncon0 6169  ‘cfv 6335  ωcom 7579  1oc1o 8105   ≈ cen 8524  cardccrd 9397 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401 This theorem is referenced by: (None)
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