Theorem pm54.43lem 9995

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 9963), 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 9996. (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 9962	. . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
2		1onn 8639	. . . . 5 ⊢ 1o ∈ ω
3		cardnn 9958	. . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o)
4	2, 3	ax-mp 5	. . . 4 ⊢ (card‘1o) = 1o
5	1, 4	eqtrdi 2789	. . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o)
6	4	eqeq2i 2746	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
7	6	biimpri 227	. . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o))
8		1n0 8488	. . . . . . . 8 ⊢ 1o ≠ ∅
9	8	neii 2943	. . . . . . 7 ⊢ ¬ 1o = ∅
10		eqeq1 2737	. . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅))
11	9, 10	mtbiri 327	. . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅)
12		ndmfv 6927	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
13	11, 12	nsyl2 141	. . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card)
14		1on 8478	. . . . . 6 ⊢ 1o ∈ On
15		onenon 9944	. . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 1o ∈ dom card
17		carden2 9982	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
18	13, 16, 17	sylancl 587	. . . 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 6901	. . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o))
22	13, 21	elab3 3677	. 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 205   = wceq 1542   ∈ wcel 2107  {cab 2710  ∅c0 4323   class class class wbr 5149  dom cdm 5677  Oncon0 6365  ‘cfv 6544  ωcom 7855  1_oc1o 8459   ≈ cen 8936  cardccrd 9930

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934

This theorem is referenced by: (None)