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Theorem pm54.43lem 9417
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 9386), 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 9418. (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 9385 . . . 4 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2 1onn 8255 . . . . 5 1o ∈ ω
3 cardnn 9381 . . . . 5 (1o ∈ ω → (card‘1o) = 1o)
42, 3ax-mp 5 . . . 4 (card‘1o) = 1o
51, 4syl6eq 2877 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = 1o)
64eqeq2i 2839 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
76biimpri 229 . . . 4 ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8 1n0 8110 . . . . . . . 8 1o ≠ ∅
98neii 3023 . . . . . . 7 ¬ 1o = ∅
10 eqeq1 2830 . . . . . . 7 ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
119, 10mtbiri 328 . . . . . 6 ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12 ndmfv 6697 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 143 . . . . 5 ((card‘𝐴) = 1o𝐴 ∈ dom card)
14 1on 8100 . . . . . 6 1o ∈ On
15 onenon 9367 . . . . . 6 (1o ∈ On → 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 9405 . . . . 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 233 . . 3 ((card‘𝐴) = 1o𝐴 ≈ 1o)
205, 19impbii 210 . 2 (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
2113elexd 3520 . . 3 ((card‘𝐴) = 1o𝐴 ∈ V)
22 fveqeq2 6676 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
2321, 22elab3 3678 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
2420, 23bitr4i 279 1 (𝐴 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2107  {cab 2804  c0 4295   class class class wbr 5063  dom cdm 5554  Oncon0 6189  cfv 6352  ωcom 7568  1oc1o 8086  cen 8495  cardccrd 9353
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1o 8093  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357
This theorem is referenced by: (None)
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