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Theorem pm54.43lem 9025
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 8994), 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 9026. (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 8993 . . . 4 (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜))
2 1onn 7873 . . . . 5 1𝑜 ∈ ω
3 cardnn 8989 . . . . 5 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . 4 (card‘1𝑜) = 1𝑜
51, 4syl6eq 2821 . . 3 (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜)
64eqeq2i 2783 . . . . 5 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
76biimpri 218 . . . 4 ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜))
8 1n0 7729 . . . . . . . 8 1𝑜 ≠ ∅
98neii 2945 . . . . . . 7 ¬ 1𝑜 = ∅
10 eqeq1 2775 . . . . . . 7 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅))
119, 10mtbiri 316 . . . . . 6 ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅)
12 ndmfv 6359 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 144 . . . . 5 ((card‘𝐴) = 1𝑜𝐴 ∈ dom card)
14 1on 7720 . . . . . 6 1𝑜 ∈ On
15 onenon 8975 . . . . . 6 (1𝑜 ∈ On → 1𝑜 ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1𝑜 ∈ dom card
17 carden2 9013 . . . . 5 ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
1813, 16, 17sylancl 574 . . . 4 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
197, 18mpbid 222 . . 3 ((card‘𝐴) = 1𝑜𝐴 ≈ 1𝑜)
205, 19impbii 199 . 2 (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜)
21 elex 3364 . . . 4 (𝐴 ∈ dom card → 𝐴 ∈ V)
2213, 21syl 17 . . 3 ((card‘𝐴) = 1𝑜𝐴 ∈ V)
23 fveq2 6332 . . . 4 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
2423eqeq1d 2773 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜))
2522, 24elab3 3509 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜)
2620, 25bitr4i 267 1 (𝐴 ≈ 1𝑜𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  c0 4063   class class class wbr 4786  dom cdm 5249  Oncon0 5866  cfv 6031  ωcom 7212  1𝑜c1o 7706  cen 8106  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-1o 7713  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965
This theorem is referenced by: (None)
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