Theorem pm54.43lem 9953

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 9921), 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 9954. (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 9920	. . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2		1onn 8604	. . . . 5 ⊢ 1o ∈ ω
3		cardnn 9916	. . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1o) = 1o
5	1, 4	eqtrdi 2780	. . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o)
6	4	eqeq2i 2742	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
7	6	biimpri 228	. . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8		1n0 8452	. . . . . . . 8 ⊢ 1o ≠ ∅
9	8	neii 2927	. . . . . . 7 ⊢ ¬ 1o = ∅
10		eqeq1 2733	. . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
11	9, 10	mtbiri 327	. . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12		ndmfv 6893	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 141	. . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card)
14		1on 8446	. . . . . 6 ⊢ 1o ∈ On
15		onenon 9902	. . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1o ∈ dom card
17		carden2 9940	. . . . 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 6867	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
22	13, 21	elab3 3653	. 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 2109  {cab 2707  ∅c0 4296   class class class wbr 5107  dom cdm 5638  Oncon0 6332  ‘cfv 6511  ωcom 7842  1_oc1o 8427   ≈ cen 8915  cardccrd 9888

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892

This theorem is referenced by: (None)