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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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