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Theorem card2inf 8997
Description: The alternate definition of the cardinal of a set given in cardval2 9398 has the curious property that for non-numerable sets (for which ndmfv 6676 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 5045 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ≺ 𝐴))
2 breq1 5045 . . . . 5 (𝑥 = 𝑛 → (𝑥𝐴𝑛𝐴))
3 breq1 5045 . . . . 5 (𝑥 = suc 𝑛 → (𝑥𝐴 ↔ suc 𝑛𝐴))
4 0elon 6220 . . . . . . . 8 ∅ ∈ On
5 breq1 5045 . . . . . . . . 9 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ≈ 𝐴))
65rspcev 3602 . . . . . . . 8 ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
74, 6mpan 688 . . . . . . 7 (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦𝐴)
87con3i 157 . . . . . 6 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ¬ ∅ ≈ 𝐴)
9 card2inf.1 . . . . . . . 8 𝐴 ∈ V
1090dom 8625 . . . . . . 7 ∅ ≼ 𝐴
11 brsdom 8510 . . . . . . 7 (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
1210, 11mpbiran 707 . . . . . 6 (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
138, 12sylibr 236 . . . . 5 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∅ ≺ 𝐴)
14 sucdom2 8692 . . . . . . . 8 (𝑛𝐴 → suc 𝑛𝐴)
1514ad2antll 727 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
16 nnon 7564 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ∈ On)
17 suceloni 7506 . . . . . . . . . 10 (𝑛 ∈ On → suc 𝑛 ∈ On)
18 breq1 5045 . . . . . . . . . . . 12 (𝑦 = suc 𝑛 → (𝑦𝐴 ↔ suc 𝑛𝐴))
1918rspcev 3602 . . . . . . . . . . 11 ((suc 𝑛 ∈ On ∧ suc 𝑛𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
2019ex 415 . . . . . . . . . 10 (suc 𝑛 ∈ On → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2116, 17, 203syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2221con3dimp 411 . . . . . . . 8 ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦𝐴) → ¬ suc 𝑛𝐴)
2322adantrr 715 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → ¬ suc 𝑛𝐴)
24 brsdom 8510 . . . . . . 7 (suc 𝑛𝐴 ↔ (suc 𝑛𝐴 ∧ ¬ suc 𝑛𝐴))
2515, 23, 24sylanbrc 585 . . . . . 6 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
2625exp32 423 . . . . 5 (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑛𝐴 → suc 𝑛𝐴)))
271, 2, 3, 13, 26finds2 7588 . . . 4 (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴𝑥𝐴))
2827com12 32 . . 3 (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑥 ∈ ω → 𝑥𝐴))
2928ralrimiv 3168 . 2 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∀𝑥 ∈ ω 𝑥𝐴)
30 omsson 7562 . . 3 ω ⊆ On
31 ssrab 4028 . . 3 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥𝐴))
3230, 31mpbiran 707 . 2 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑥 ∈ ω 𝑥𝐴)
3329, 32sylibr 236 1 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wcel 2114  wral 3125  wrex 3126  {crab 3129  Vcvv 3473  wss 3913  c0 4269   class class class wbr 5042  Oncon0 6167  suc csuc 6169  ωcom 7558  cen 8484  cdom 8485  csdm 8486
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 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
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 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-om 7559  df-1o 8080  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490
This theorem is referenced by: (None)
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