Theorem card2inf 9588

Description: The alternate definition of the cardinal of a set given in cardval2 10024 has the curious property that for non-numerable sets (for which ndmfv 6937 yields ∅), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Hypothesis

Ref	Expression
card2inf.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
card2inf	⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem card2inf

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5155	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
2		breq1 5155	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3		breq1 5155	. . . . 5 ⊢ (𝑥 = suc 𝑛 → (𝑥 ≺ 𝐴 ↔ suc 𝑛 ≺ 𝐴))
4		0elon 6428	. . . . . . . 8 ⊢ ∅ ∈ On
5		breq1 5155	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 ≈ 𝐴 ↔ ∅ ≈ 𝐴))
6	5	rspcev 3611	. . . . . . . 8 ⊢ ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
7	4, 6	mpan 688	. . . . . . 7 ⊢ (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
8	7	con3i 154	. . . . . 6 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ¬ ∅ ≈ 𝐴)
9		card2inf.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
10	9	0dom 9139	. . . . . . 7 ⊢ ∅ ≼ 𝐴
11		brsdom 9004	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
12	10, 11	mpbiran 707	. . . . . 6 ⊢ (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
13	8, 12	sylibr 233	. . . . 5 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∅ ≺ 𝐴)
14		sucdom2 9239	. . . . . . . 8 ⊢ (𝑛 ≺ 𝐴 → suc 𝑛 ≼ 𝐴)
15	14	ad2antll 727	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≼ 𝐴)
16		nnon 7884	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
17		onsuc 7822	. . . . . . . . . 10 ⊢ (𝑛 ∈ On → suc 𝑛 ∈ On)
18		breq1 5155	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑛 → (𝑦 ≈ 𝐴 ↔ suc 𝑛 ≈ 𝐴))
19	18	rspcev 3611	. . . . . . . . . . 11 ⊢ ((suc 𝑛 ∈ On ∧ suc 𝑛 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
20	19	ex 411	. . . . . . . . . 10 ⊢ (suc 𝑛 ∈ On → (suc 𝑛 ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴))
21	16, 17, 20	3syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (suc 𝑛 ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴))
22	21	con3dimp 407	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴) → ¬ suc 𝑛 ≈ 𝐴)
23	22	adantrr 715	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → ¬ suc 𝑛 ≈ 𝐴)
24		brsdom 9004	. . . . . . 7 ⊢ (suc 𝑛 ≺ 𝐴 ↔ (suc 𝑛 ≼ 𝐴 ∧ ¬ suc 𝑛 ≈ 𝐴))
25	15, 23, 24	sylanbrc 581	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≺ 𝐴)
26	25	exp32 419	. . . . 5 ⊢ (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑛 ≺ 𝐴 → suc 𝑛 ≺ 𝐴)))
27	1, 2, 3, 13, 26	finds2 7914	. . . 4 ⊢ (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝑥 ≺ 𝐴))
28	27	com12 32	. . 3 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑥 ∈ ω → 𝑥 ≺ 𝐴))
29	28	ralrimiv 3142	. 2 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
30		omsson 7882	. . 3 ⊢ ω ⊆ On
31		ssrab 4070	. . 3 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴))
32	30, 31	mpbiran 707	. 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
33	29, 32	sylibr 233	1 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  {crab 3430  Vcvv 3473   ⊆ wss 3949  ∅c0 4326   class class class wbr 5152  Oncon0 6374  suc csuc 6376  ωcom 7878   ≈ cen 8969   ≼ cdom 8970   ≺ csdm 8971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7879  df-1o 8495  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976

This theorem is referenced by: (None)