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Theorem pm54.43lem 8685
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 8654), 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 8686. (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 8653 . . . 4 (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜))
2 1onn 7583 . . . . 5 1𝑜 ∈ ω
3 cardnn 8649 . . . . 5 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . 4 (card‘1𝑜) = 1𝑜
51, 4syl6eq 2659 . . 3 (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜)
64eqeq2i 2621 . . . . 5 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
76biimpri 216 . . . 4 ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜))
8 1n0 7439 . . . . . . . 8 1𝑜 ≠ ∅
98neii 2783 . . . . . . 7 ¬ 1𝑜 = ∅
10 eqeq1 2613 . . . . . . 7 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅))
119, 10mtbiri 315 . . . . . 6 ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅)
12 ndmfv 6113 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 140 . . . . 5 ((card‘𝐴) = 1𝑜𝐴 ∈ dom card)
14 1on 7431 . . . . . 6 1𝑜 ∈ On
15 onenon 8635 . . . . . 6 (1𝑜 ∈ On → 1𝑜 ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1𝑜 ∈ dom card
17 carden2 8673 . . . . 5 ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
1813, 16, 17sylancl 692 . . . 4 ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
197, 18mpbid 220 . . 3 ((card‘𝐴) = 1𝑜𝐴 ≈ 1𝑜)
205, 19impbii 197 . 2 (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜)
21 elex 3184 . . . 4 (𝐴 ∈ dom card → 𝐴 ∈ V)
2213, 21syl 17 . . 3 ((card‘𝐴) = 1𝑜𝐴 ∈ V)
23 fveq2 6088 . . . 4 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
2423eqeq1d 2611 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜))
2522, 24elab3 3326 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜)
2620, 25bitr4i 265 1 (𝐴 ≈ 1𝑜𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  {cab 2595  Vcvv 3172  c0 3873   class class class wbr 4577  dom cdm 5028  Oncon0 5626  cfv 5790  ωcom 6934  1𝑜c1o 7417  cen 7815  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-1o 7424  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625
This theorem is referenced by: (None)
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