Theorem pm54.43lem 8863

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 8832), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1_𝑜}. Here we show that this is equivalent to 𝐴 ≈ 1_𝑜 so that we can use the latter more convenient notation in pm54.43 8864. (Contributed by NM, 4-Nov-2013.)

Assertion

Ref	Expression
pm54.43lem	⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})

Distinct variable group:   𝑥,𝐴

Proof of Theorem pm54.43lem

Step	Hyp	Ref	Expression
1		carden2b 8831	. . . 4 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜))
2		1onn 7764	. . . . 5 ⊢ 1𝑜 ∈ ω
3		cardnn 8827	. . . . 5 ⊢ (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1𝑜) = 1𝑜
5	1, 4	syl6eq 2701	. . 3 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜)
6	4	eqeq2i 2663	. . . . 5 ⊢ ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
7	6	biimpri 218	. . . 4 ⊢ ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜))
8		1n0 7620	. . . . . . . 8 ⊢ 1𝑜 ≠ ∅
9	8	neii 2825	. . . . . . 7 ⊢ ¬ 1𝑜 = ∅
10		eqeq1 2655	. . . . . . 7 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅))
11	9, 10	mtbiri 316	. . . . . 6 ⊢ ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅)
12		ndmfv 6256	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 142	. . . . 5 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ dom card)
14		1on 7612	. . . . . 6 ⊢ 1𝑜 ∈ On
15		onenon 8813	. . . . . 6 ⊢ (1𝑜 ∈ On → 1𝑜 ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1𝑜 ∈ dom card
17		carden2 8851	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
18	13, 16, 17	sylancl 695	. . . 4 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
19	7, 18	mpbid 222	. . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ≈ 1𝑜)
20	5, 19	impbii 199	. 2 ⊢ (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜)
21		elex 3243	. . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V)
22	13, 21	syl 17	. . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ V)
23		fveq2 6229	. . . 4 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
24	23	eqeq1d 2653	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜))
25	22, 24	elab3 3390	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜)
26	20, 25	bitr4i 267	1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})

Syntax hints:   ↔ wb 196   = wceq 1523   ∈ wcel 2030  {cab 2637  Vcvv 3231  ∅c0 3948   class class class wbr 4685  dom cdm 5143  Oncon0 5761  ‘cfv 5926  ωcom 7107  1_𝑜c1o 7598   ≈ cen 7994  cardccrd 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803

This theorem is referenced by: (None)