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Theorem card2inf 8749
Description: The alternate definition of the cardinal of a set given in cardval2 9150 has the curious property that for non-numerable sets (for which ndmfv 6476 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 4889 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ≺ 𝐴))
2 breq1 4889 . . . . 5 (𝑥 = 𝑛 → (𝑥𝐴𝑛𝐴))
3 breq1 4889 . . . . 5 (𝑥 = suc 𝑛 → (𝑥𝐴 ↔ suc 𝑛𝐴))
4 0elon 6029 . . . . . . . 8 ∅ ∈ On
5 breq1 4889 . . . . . . . . 9 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ≈ 𝐴))
65rspcev 3511 . . . . . . . 8 ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
74, 6mpan 680 . . . . . . 7 (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦𝐴)
87con3i 152 . . . . . 6 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ¬ ∅ ≈ 𝐴)
9 card2inf.1 . . . . . . . 8 𝐴 ∈ V
1090dom 8378 . . . . . . 7 ∅ ≼ 𝐴
11 brsdom 8264 . . . . . . 7 (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
1210, 11mpbiran 699 . . . . . 6 (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
138, 12sylibr 226 . . . . 5 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∅ ≺ 𝐴)
14 sucdom2 8444 . . . . . . . 8 (𝑛𝐴 → suc 𝑛𝐴)
1514ad2antll 719 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
16 nnon 7349 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ∈ On)
17 suceloni 7291 . . . . . . . . . 10 (𝑛 ∈ On → suc 𝑛 ∈ On)
18 breq1 4889 . . . . . . . . . . . 12 (𝑦 = suc 𝑛 → (𝑦𝐴 ↔ suc 𝑛𝐴))
1918rspcev 3511 . . . . . . . . . . 11 ((suc 𝑛 ∈ On ∧ suc 𝑛𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
2019ex 403 . . . . . . . . . 10 (suc 𝑛 ∈ On → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2116, 17, 203syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2221con3dimp 399 . . . . . . . 8 ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦𝐴) → ¬ suc 𝑛𝐴)
2322adantrr 707 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → ¬ suc 𝑛𝐴)
24 brsdom 8264 . . . . . . 7 (suc 𝑛𝐴 ↔ (suc 𝑛𝐴 ∧ ¬ suc 𝑛𝐴))
2515, 23, 24sylanbrc 578 . . . . . 6 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
2625exp32 413 . . . . 5 (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑛𝐴 → suc 𝑛𝐴)))
271, 2, 3, 13, 26finds2 7372 . . . 4 (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴𝑥𝐴))
2827com12 32 . . 3 (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑥 ∈ ω → 𝑥𝐴))
2928ralrimiv 3147 . 2 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∀𝑥 ∈ ω 𝑥𝐴)
30 omsson 7347 . . 3 ω ⊆ On
31 ssrab 3901 . . 3 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥𝐴))
3230, 31mpbiran 699 . 2 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑥 ∈ ω 𝑥𝐴)
3329, 32sylibr 226 1 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wcel 2107  wral 3090  wrex 3091  {crab 3094  Vcvv 3398  wss 3792  c0 4141   class class class wbr 4886  Oncon0 5976  suc csuc 5978  ωcom 7343  cen 8238  cdom 8239  csdm 8240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-om 7344  df-1o 7843  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244
This theorem is referenced by: (None)
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