Theorem pm54.43lem 9758

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 9726), 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 9759. (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 9725	. . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2		1onn 8470	. . . . 5 ⊢ 1o ∈ ω
3		cardnn 9721	. . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1o) = 1o
5	1, 4	eqtrdi 2794	. . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o)
6	4	eqeq2i 2751	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
7	6	biimpri 227	. . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8		1n0 8318	. . . . . . . 8 ⊢ 1o ≠ ∅
9	8	neii 2945	. . . . . . 7 ⊢ ¬ 1o = ∅
10		eqeq1 2742	. . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
11	9, 10	mtbiri 327	. . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12		ndmfv 6804	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 141	. . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card)
14		1on 8309	. . . . . 6 ⊢ 1o ∈ On
15		onenon 9707	. . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1o ∈ dom card
17		carden2 9745	. . . . 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 231	. . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o)
20	5, 19	impbii 208	. 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o)
21		fveqeq2 6783	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
22	13, 21	elab3 3617	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o)
23	20, 22	bitr4i 277	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  ∅c0 4256   class class class wbr 5074  dom cdm 5589  Oncon0 6266  ‘cfv 6433  ωcom 7712  1_oc1o 8290   ≈ cen 8730  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697

This theorem is referenced by: (None)