Theorem pm54.43lem 10014

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 9982), 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 10015. (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 9981	. . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2		1onn 8652	. . . . 5 ⊢ 1o ∈ ω
3		cardnn 9977	. . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1o) = 1o
5	1, 4	eqtrdi 2786	. . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o)
6	4	eqeq2i 2748	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
7	6	biimpri 228	. . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8		1n0 8500	. . . . . . . 8 ⊢ 1o ≠ ∅
9	8	neii 2934	. . . . . . 7 ⊢ ¬ 1o = ∅
10		eqeq1 2739	. . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
11	9, 10	mtbiri 327	. . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12		ndmfv 6911	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 141	. . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card)
14		1on 8492	. . . . . 6 ⊢ 1o ∈ On
15		onenon 9963	. . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1o ∈ dom card
17		carden2 10001	. . . . 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 6885	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
22	13, 21	elab3 3665	. 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 2713  ∅c0 4308   class class class wbr 5119  dom cdm 5654  Oncon0 6352  ‘cfv 6531  ωcom 7861  1_oc1o 8473   ≈ cen 8956  cardccrd 9949

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953

This theorem is referenced by: (None)