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Theorem pm54.43lem 10038
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 10006), 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 10039. (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 10005 . . . 4 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2 1onn 8677 . . . . 5 1o ∈ ω
3 cardnn 10001 . . . . 5 (1o ∈ ω → (card‘1o) = 1o)
42, 3ax-mp 5 . . . 4 (card‘1o) = 1o
51, 4eqtrdi 2791 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = 1o)
64eqeq2i 2748 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
76biimpri 228 . . . 4 ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8 1n0 8525 . . . . . . . 8 1o ≠ ∅
98neii 2940 . . . . . . 7 ¬ 1o = ∅
10 eqeq1 2739 . . . . . . 7 ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
119, 10mtbiri 327 . . . . . 6 ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12 ndmfv 6942 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 141 . . . . 5 ((card‘𝐴) = 1o𝐴 ∈ dom card)
14 1on 8517 . . . . . 6 1o ∈ On
15 onenon 9987 . . . . . 6 (1o ∈ On → 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 10025 . . . . 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 6916 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
2213, 21elab3 3689 . 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 1537  wcel 2106  {cab 2712  c0 4339   class class class wbr 5148  dom cdm 5689  Oncon0 6386  cfv 6563  ωcom 7887  1oc1o 8498  cen 8981  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977
This theorem is referenced by: (None)
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