Theorem card2inf 9460

Description: The alternate definition of the cardinal of a set given in cardval2 9903 has the curious property that for non-numerable sets (for which ndmfv 6866 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 5101	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
2		breq1 5101	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3		breq1 5101	. . . . 5 ⊢ (𝑥 = suc 𝑛 → (𝑥 ≺ 𝐴 ↔ suc 𝑛 ≺ 𝐴))
4		0elon 6372	. . . . . . . 8 ⊢ ∅ ∈ On
5		breq1 5101	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 ≈ 𝐴 ↔ ∅ ≈ 𝐴))
6	5	rspcev 3576	. . . . . . . 8 ⊢ ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
7	4, 6	mpan 690	. . . . . . 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 9035	. . . . . . 7 ⊢ ∅ ≼ 𝐴
11		brsdom 8911	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
12	10, 11	mpbiran 709	. . . . . 6 ⊢ (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
13	8, 12	sylibr 234	. . . . 5 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∅ ≺ 𝐴)
14		sucdom2 9127	. . . . . . . 8 ⊢ (𝑛 ≺ 𝐴 → suc 𝑛 ≼ 𝐴)
15	14	ad2antll 729	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≼ 𝐴)
16		nnon 7814	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
17		onsuc 7755	. . . . . . . . . 10 ⊢ (𝑛 ∈ On → suc 𝑛 ∈ On)
18		breq1 5101	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑛 → (𝑦 ≈ 𝐴 ↔ suc 𝑛 ≈ 𝐴))
19	18	rspcev 3576	. . . . . . . . . . 11 ⊢ ((suc 𝑛 ∈ On ∧ suc 𝑛 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
20	19	ex 412	. . . . . . . . . 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 408	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴) → ¬ suc 𝑛 ≈ 𝐴)
23	22	adantrr 717	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → ¬ suc 𝑛 ≈ 𝐴)
24		brsdom 8911	. . . . . . 7 ⊢ (suc 𝑛 ≺ 𝐴 ↔ (suc 𝑛 ≼ 𝐴 ∧ ¬ suc 𝑛 ≈ 𝐴))
25	15, 23, 24	sylanbrc 583	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≺ 𝐴)
26	25	exp32 420	. . . . 5 ⊢ (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑛 ≺ 𝐴 → suc 𝑛 ≺ 𝐴)))
27	1, 2, 3, 13, 26	finds2 7840	. . . 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 3127	. 2 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
30		omsson 7812	. . 3 ⊢ ω ⊆ On
31		ssrab 4023	. . 3 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴))
32	30, 31	mpbiran 709	. 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
33	29, 32	sylibr 234	1 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  Oncon0 6317  suc csuc 6319  ωcom 7808   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887

This theorem is referenced by: (None)