Theorem card2inf 9021

Description: The alternate definition of the cardinal of a set given in cardval2 9422 has the curious property that for non-numerable sets (for which ndmfv 6702 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 5071	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
2		breq1 5071	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3		breq1 5071	. . . . 5 ⊢ (𝑥 = suc 𝑛 → (𝑥 ≺ 𝐴 ↔ suc 𝑛 ≺ 𝐴))
4		0elon 6246	. . . . . . . 8 ⊢ ∅ ∈ On
5		breq1 5071	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 ≈ 𝐴 ↔ ∅ ≈ 𝐴))
6	5	rspcev 3625	. . . . . . . 8 ⊢ ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
7	4, 6	mpan 688	. . . . . . 7 ⊢ (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
8	7	con3i 157	. . . . . 6 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ¬ ∅ ≈ 𝐴)
9		card2inf.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
10	9	0dom 8649	. . . . . . 7 ⊢ ∅ ≼ 𝐴
11		brsdom 8534	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
12	10, 11	mpbiran 707	. . . . . 6 ⊢ (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
13	8, 12	sylibr 236	. . . . 5 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∅ ≺ 𝐴)
14		sucdom2 8716	. . . . . . . 8 ⊢ (𝑛 ≺ 𝐴 → suc 𝑛 ≼ 𝐴)
15	14	ad2antll 727	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≼ 𝐴)
16		nnon 7588	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
17		suceloni 7530	. . . . . . . . . 10 ⊢ (𝑛 ∈ On → suc 𝑛 ∈ On)
18		breq1 5071	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑛 → (𝑦 ≈ 𝐴 ↔ suc 𝑛 ≈ 𝐴))
19	18	rspcev 3625	. . . . . . . . . . 11 ⊢ ((suc 𝑛 ∈ On ∧ suc 𝑛 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
20	19	ex 415	. . . . . . . . . 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 411	. . . . . . . 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 8534	. . . . . . 7 ⊢ (suc 𝑛 ≺ 𝐴 ↔ (suc 𝑛 ≼ 𝐴 ∧ ¬ suc 𝑛 ≈ 𝐴))
25	15, 23, 24	sylanbrc 585	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≺ 𝐴)
26	25	exp32 423	. . . . 5 ⊢ (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑛 ≺ 𝐴 → suc 𝑛 ≺ 𝐴)))
27	1, 2, 3, 13, 26	finds2 7612	. . . 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 3183	. 2 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
30		omsson 7586	. . 3 ⊢ ω ⊆ On
31		ssrab 4051	. . 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 236	1 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068  Oncon0 6193  suc csuc 6195  ωcom 7582   ≈ cen 8508   ≼ cdom 8509   ≺ csdm 8510

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-om 7583  df-1o 8104  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514

This theorem is referenced by: (None)