Theorem card2inf 9500

Description: The alternate definition of the cardinal of a set given in cardval2 9946 has the curious property that for non-numerable sets (for which ndmfv 6895 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 5102	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
2		breq1 5102	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3		breq1 5102	. . . . 5 ⊢ (𝑥 = suc 𝑛 → (𝑥 ≺ 𝐴 ↔ suc 𝑛 ≺ 𝐴))
4		0elon 6397	. . . . . . . 8 ⊢ ∅ ∈ On
5		breq1 5102	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 ≈ 𝐴 ↔ ∅ ≈ 𝐴))
6	5	rspcev 3581	. . . . . . . 8 ⊢ ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
7	4, 6	mpan 700	. . . . . . 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 9075	. . . . . . 7 ⊢ ∅ ≼ 𝐴
11		brsdom 8951	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
12	10, 11	mpbiran 719	. . . . . 6 ⊢ (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
13	8, 12	sylibr 236	. . . . 5 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∅ ≺ 𝐴)
14		sucdom2 9167	. . . . . . . 8 ⊢ (𝑛 ≺ 𝐴 → suc 𝑛 ≼ 𝐴)
15	14	ad2antll 739	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≼ 𝐴)
16		nnon 7848	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
17		onsuc 7789	. . . . . . . . . 10 ⊢ (𝑛 ∈ On → suc 𝑛 ∈ On)
18		breq1 5102	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑛 → (𝑦 ≈ 𝐴 ↔ suc 𝑛 ≈ 𝐴))
19	18	rspcev 3581	. . . . . . . . . . 11 ⊢ ((suc 𝑛 ∈ On ∧ suc 𝑛 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
20	19	ex 416	. . . . . . . . . 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 412	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴) → ¬ suc 𝑛 ≈ 𝐴)
23	22	adantrr 727	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → ¬ suc 𝑛 ≈ 𝐴)
24		brsdom 8951	. . . . . . 7 ⊢ (suc 𝑛 ≺ 𝐴 ↔ (suc 𝑛 ≼ 𝐴 ∧ ¬ suc 𝑛 ≈ 𝐴))
25	15, 23, 24	sylanbrc 592	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≺ 𝐴)
26	25	exp32 424	. . . . 5 ⊢ (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑛 ≺ 𝐴 → suc 𝑛 ≺ 𝐴)))
27	1, 2, 3, 13, 26	finds2 7875	. . . 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 3152	. 2 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)
30		omsson 7846	. . 3 ⊢ ω ⊆ On
31		ssrab 4024	. . 3 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴))
32	30, 31	mpbiran 719	. 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 399   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453   ⊆ wss 3904  ∅c0 4285   class class class wbr 5099  Oncon0 6342  suc csuc 6344  ωcom 7842   ≈ cen 8920   ≼ cdom 8921   ≺ csdm 8922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-om 7843  df-1o 8432  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927

This theorem is referenced by: (None)