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Theorem card2inf 9498
Description: The alternate definition of the cardinal of a set given in cardval2 9934 has the curious property that for non-numerable sets (for which ndmfv 6882 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 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem card2inf
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 breq1 5113 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ≺ 𝐴))
2 breq1 5113 . . . . 5 (𝑥 = 𝑛 → (𝑥𝐴𝑛𝐴))
3 breq1 5113 . . . . 5 (𝑥 = suc 𝑛 → (𝑥𝐴 ↔ suc 𝑛𝐴))
4 0elon 6376 . . . . . . . 8 ∅ ∈ On
5 breq1 5113 . . . . . . . . 9 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ≈ 𝐴))
65rspcev 3584 . . . . . . . 8 ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
74, 6mpan 689 . . . . . . 7 (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦𝐴)
87con3i 154 . . . . . 6 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ¬ ∅ ≈ 𝐴)
9 card2inf.1 . . . . . . . 8 𝐴 ∈ V
1090dom 9057 . . . . . . 7 ∅ ≼ 𝐴
11 brsdom 8922 . . . . . . 7 (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
1210, 11mpbiran 708 . . . . . 6 (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
138, 12sylibr 233 . . . . 5 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∅ ≺ 𝐴)
14 sucdom2 9157 . . . . . . . 8 (𝑛𝐴 → suc 𝑛𝐴)
1514ad2antll 728 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
16 nnon 7813 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ∈ On)
17 onsuc 7751 . . . . . . . . . 10 (𝑛 ∈ On → suc 𝑛 ∈ On)
18 breq1 5113 . . . . . . . . . . . 12 (𝑦 = suc 𝑛 → (𝑦𝐴 ↔ suc 𝑛𝐴))
1918rspcev 3584 . . . . . . . . . . 11 ((suc 𝑛 ∈ On ∧ suc 𝑛𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
2019ex 414 . . . . . . . . . 10 (suc 𝑛 ∈ On → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2116, 17, 203syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2221con3dimp 410 . . . . . . . 8 ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦𝐴) → ¬ suc 𝑛𝐴)
2322adantrr 716 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → ¬ suc 𝑛𝐴)
24 brsdom 8922 . . . . . . 7 (suc 𝑛𝐴 ↔ (suc 𝑛𝐴 ∧ ¬ suc 𝑛𝐴))
2515, 23, 24sylanbrc 584 . . . . . 6 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
2625exp32 422 . . . . 5 (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑛𝐴 → suc 𝑛𝐴)))
271, 2, 3, 13, 26finds2 7842 . . . 4 (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴𝑥𝐴))
2827com12 32 . . 3 (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑥 ∈ ω → 𝑥𝐴))
2928ralrimiv 3143 . 2 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∀𝑥 ∈ ω 𝑥𝐴)
30 omsson 7811 . . 3 ω ⊆ On
31 ssrab 4035 . . 3 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥𝐴))
3230, 31mpbiran 708 . 2 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑥 ∈ ω 𝑥𝐴)
3329, 32sylibr 233 1 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  wral 3065  wrex 3074  {crab 3410  Vcvv 3448  wss 3915  c0 4287   class class class wbr 5110  Oncon0 6322  suc csuc 6324  ωcom 7807  cen 8887  cdom 8888  csdm 8889
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by: (None)
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