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Theorem pm54.43lem 9145
 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 9114), 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 9146. (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 9113 . . . 4 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2 1onn 7991 . . . . 5 1o ∈ ω
3 cardnn 9109 . . . . 5 (1o ∈ ω → (card‘1o) = 1o)
42, 3ax-mp 5 . . . 4 (card‘1o) = 1o
51, 4syl6eq 2877 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = 1o)
64eqeq2i 2837 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
76biimpri 220 . . . 4 ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8 1n0 7847 . . . . . . . 8 1o ≠ ∅
98neii 3001 . . . . . . 7 ¬ 1o = ∅
10 eqeq1 2829 . . . . . . 7 ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
119, 10mtbiri 319 . . . . . 6 ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12 ndmfv 6467 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1311, 12nsyl2 145 . . . . 5 ((card‘𝐴) = 1o𝐴 ∈ dom card)
14 1on 7838 . . . . . 6 1o ∈ On
15 onenon 9095 . . . . . 6 (1o ∈ On → 1o ∈ dom card)
1614, 15ax-mp 5 . . . . 5 1o ∈ dom card
17 carden2 9133 . . . . 5 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
1813, 16, 17sylancl 580 . . . 4 ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
197, 18mpbid 224 . . 3 ((card‘𝐴) = 1o𝐴 ≈ 1o)
205, 19impbii 201 . 2 (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
21 elex 3429 . . . 4 (𝐴 ∈ dom card → 𝐴 ∈ V)
2213, 21syl 17 . . 3 ((card‘𝐴) = 1o𝐴 ∈ V)
23 fveqeq2 6446 . . 3 (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
2422, 23elab3 3579 . 2 (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
2520, 24bitr4i 270 1 (𝐴 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1656   ∈ wcel 2164  {cab 2811  Vcvv 3414  ∅c0 4146   class class class wbr 4875  dom cdm 5346  Oncon0 5967  ‘cfv 6127  ωcom 7331  1oc1o 7824   ≈ cen 8225  cardccrd 9081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-1o 7831  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085 This theorem is referenced by: (None)
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