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Theorem card2inf 8411
 Description: The definition cardval2 8768 has the curious property that for non-numerable sets (for which ndmfv 6180 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 4621 . . . . 5 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ≺ 𝐴))
2 breq1 4621 . . . . 5 (𝑥 = 𝑛 → (𝑥𝐴𝑛𝐴))
3 breq1 4621 . . . . 5 (𝑥 = suc 𝑛 → (𝑥𝐴 ↔ suc 𝑛𝐴))
4 0elon 5742 . . . . . . . 8 ∅ ∈ On
5 breq1 4621 . . . . . . . . 9 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ≈ 𝐴))
65rspcev 3298 . . . . . . . 8 ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
74, 6mpan 705 . . . . . . 7 (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦𝐴)
87con3i 150 . . . . . 6 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ¬ ∅ ≈ 𝐴)
9 card2inf.1 . . . . . . . 8 𝐴 ∈ V
1090dom 8041 . . . . . . 7 ∅ ≼ 𝐴
11 brsdom 7929 . . . . . . 7 (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴))
1210, 11mpbiran 952 . . . . . 6 (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴)
138, 12sylibr 224 . . . . 5 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∅ ≺ 𝐴)
14 sucdom2 8107 . . . . . . . 8 (𝑛𝐴 → suc 𝑛𝐴)
1514ad2antll 764 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
16 nnon 7025 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ∈ On)
17 suceloni 6967 . . . . . . . . . 10 (𝑛 ∈ On → suc 𝑛 ∈ On)
18 breq1 4621 . . . . . . . . . . . 12 (𝑦 = suc 𝑛 → (𝑦𝐴 ↔ suc 𝑛𝐴))
1918rspcev 3298 . . . . . . . . . . 11 ((suc 𝑛 ∈ On ∧ suc 𝑛𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
2019ex 450 . . . . . . . . . 10 (suc 𝑛 ∈ On → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2116, 17, 203syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛𝐴 → ∃𝑦 ∈ On 𝑦𝐴))
2221con3dimp 457 . . . . . . . 8 ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦𝐴) → ¬ suc 𝑛𝐴)
2322adantrr 752 . . . . . . 7 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → ¬ suc 𝑛𝐴)
24 brsdom 7929 . . . . . . 7 (suc 𝑛𝐴 ↔ (suc 𝑛𝐴 ∧ ¬ suc 𝑛𝐴))
2515, 23, 24sylanbrc 697 . . . . . 6 ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦𝐴𝑛𝐴)) → suc 𝑛𝐴)
2625exp32 630 . . . . 5 (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑛𝐴 → suc 𝑛𝐴)))
271, 2, 3, 13, 26finds2 7048 . . . 4 (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦𝐴𝑥𝐴))
2827com12 32 . . 3 (¬ ∃𝑦 ∈ On 𝑦𝐴 → (𝑥 ∈ ω → 𝑥𝐴))
2928ralrimiv 2960 . 2 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ∀𝑥 ∈ ω 𝑥𝐴)
30 omsson 7023 . . 3 ω ⊆ On
31 ssrab 3664 . . 3 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥𝐴))
3230, 31mpbiran 952 . 2 (ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑥 ∈ ω 𝑥𝐴)
3329, 32sylibr 224 1 (¬ ∃𝑦 ∈ On 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911  Vcvv 3189   ⊆ wss 3559  ∅c0 3896   class class class wbr 4618  Oncon0 5687  suc csuc 5689  ωcom 7019   ≈ cen 7903   ≼ cdom 7904   ≺ csdm 7905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-om 7020  df-1o 7512  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909 This theorem is referenced by: (None)
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