Theorem pm54.43lem 10040

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 10008), 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 10041. (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 10007	. . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2		1onn 8678	. . . . 5 ⊢ 1o ∈ ω
3		cardnn 10003	. . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1o) = 1o
5	1, 4	eqtrdi 2793	. . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o)
6	4	eqeq2i 2750	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
7	6	biimpri 228	. . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8		1n0 8526	. . . . . . . 8 ⊢ 1o ≠ ∅
9	8	neii 2942	. . . . . . 7 ⊢ ¬ 1o = ∅
10		eqeq1 2741	. . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
11	9, 10	mtbiri 327	. . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12		ndmfv 6941	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 141	. . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card)
14		1on 8518	. . . . . 6 ⊢ 1o ∈ On
15		onenon 9989	. . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1o ∈ dom card
17		carden2 10027	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
18	13, 16, 17	sylancl 586	. . . 4 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
19	7, 18	mpbid 232	. . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o)
20	5, 19	impbii 209	. 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
21		fveqeq2 6915	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
22	13, 21	elab3 3686	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
23	20, 22	bitr4i 278	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {cab 2714  ∅c0 4333   class class class wbr 5143  dom cdm 5685  Oncon0 6384  ‘cfv 6561  ωcom 7887  1_oc1o 8499   ≈ cen 8982  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979

This theorem is referenced by: (None)