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Theorem pm54.43lem 8708
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 8677), 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 8709. (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 8676 . . . 4 (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜))
2 1onn 7606 . . . . 5 1𝑜 ∈ ω
3 cardnn 8672 . . . . 5 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . 4 (card‘1𝑜) = 1𝑜
51, 4syl6eq 2660 . . 3 (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜)
64eqeq2i 2622 . . . . 5 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
76biimpri 217 . . . 4 ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜))
8 1n0 7462 . . . . . . . 8 1𝑜 ≠ ∅
98neii 2784 . . . . . . 7 ¬ 1𝑜 = ∅
10 eqeq1 2614 . . . . . . 7 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅))
119, 10mtbiri 316 . . . . . 6 ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅)
12 ndmfv 6128 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 141 . . . . 5 ((card‘𝐴) = 1𝑜𝐴 ∈ dom card)
14 1on 7454 . . . . . 6 1𝑜 ∈ On
15 onenon 8658 . . . . . 6 (1𝑜 ∈ On → 1𝑜 ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1𝑜 ∈ dom card
17 carden2 8696 . . . . 5 ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
1813, 16, 17sylancl 693 . . . 4 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
197, 18mpbid 221 . . 3 ((card‘𝐴) = 1𝑜𝐴 ≈ 1𝑜)
205, 19impbii 198 . 2 (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜)
21 elex 3185 . . . 4 (𝐴 ∈ dom card → 𝐴 ∈ V)
2213, 21syl 17 . . 3 ((card‘𝐴) = 1𝑜𝐴 ∈ V)
23 fveq2 6103 . . . 4 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
2423eqeq1d 2612 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜))
2522, 24elab3 3327 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜)
2620, 25bitr4i 266 1 (𝐴 ≈ 1𝑜𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  {cab 2596  Vcvv 3173  c0 3874   class class class wbr 4583  dom cdm 5038  Oncon0 5640  cfv 5804  ωcom 6957  1𝑜c1o 7440  cen 7838  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648
This theorem is referenced by: (None)
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