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Theorem cantnflem2 8625
Description: Lemma for cantnf 8628. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
cantnf.c (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
cantnf.s (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e (𝜑 → ∅ ∈ 𝐶)
Assertion
Ref Expression
cantnflem2 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐶,𝑥,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cantnflem2
StepHypRef Expression
1 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
2 cantnfs.b . . . . . . . . . 10 (𝜑𝐵 ∈ On)
3 oecl 7662 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2anc 694 . . . . . . . . 9 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
6 onelon 5786 . . . . . . . . 9 (((𝐴𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴𝑜 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 694 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 7626 . . . . . . . 8 (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 699 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1𝑜))
1110eldifbd 3620 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12 ssel 3630 . . . . . . 7 ((𝐴𝑜 𝐵) ⊆ 1𝑜 → (𝐶 ∈ (𝐴𝑜 𝐵) → 𝐶 ∈ 1𝑜))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴𝑜 𝐵) ⊆ 1𝑜𝐶 ∈ 1𝑜))
1411, 13mtod 189 . . . . 5 (𝜑 → ¬ (𝐴𝑜 𝐵) ⊆ 1𝑜)
15 oe0m 7643 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
17 difss 3770 . . . . . . . 8 (1𝑜𝐵) ⊆ 1𝑜
1816, 17syl6eqss 3688 . . . . . . 7 (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19 oveq1 6697 . . . . . . . 8 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
2019sseq1d 3665 . . . . . . 7 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
2118, 20syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴𝑜 𝐵) ⊆ 1𝑜))
22 oe1m 7670 . . . . . . . 8 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
23 eqimss 3690 . . . . . . . 8 ((1𝑜𝑜 𝐵) = 1𝑜 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
25 oveq1 6697 . . . . . . . 8 (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) = (1𝑜𝑜 𝐵))
2625sseq1d 3665 . . . . . . 7 (𝐴 = 1𝑜 → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (1𝑜𝑜 𝐵) ⊆ 1𝑜))
2724, 26syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2821, 27jaod 394 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2914, 28mtod 189 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30 elpri 4230 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31 df2o3 7618 . . . . 5 2𝑜 = {∅, 1𝑜}
3230, 31eleq2s 2748 . . . 4 (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
3329, 32nsyl 135 . . 3 (𝜑 → ¬ 𝐴 ∈ 2𝑜)
341, 33eldifd 3618 . 2 (𝜑𝐴 ∈ (On ∖ 2𝑜))
3534, 10jca 553 1 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cdif 3604  wss 3607  c0 3948  {cpr 4212  {copab 4745  dom cdm 5143  ran crn 5144  Oncon0 5761  cfv 5926  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599  𝑜 coe 7604   CNF ccnf 8596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  cantnflem3  8626  cantnflem4  8627
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