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Theorem cantnflem2 8531
 Description: Lemma for cantnf 8534. (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 7562 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
6 onelon 5707 . . . . . . . . 9 (((𝐴𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴𝑜 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 692 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 7526 . . . . . . . 8 (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 697 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1𝑜))
1110eldifbd 3568 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12 ssel 3577 . . . . . . 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 7543 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
17 difss 3715 . . . . . . . 8 (1𝑜𝐵) ⊆ 1𝑜
1816, 17syl6eqss 3634 . . . . . . 7 (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19 oveq1 6611 . . . . . . . 8 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
2019sseq1d 3611 . . . . . . 7 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
2118, 20syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴𝑜 𝐵) ⊆ 1𝑜))
22 oe1m 7570 . . . . . . . 8 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
23 eqimss 3636 . . . . . . . 8 ((1𝑜𝑜 𝐵) = 1𝑜 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
25 oveq1 6611 . . . . . . . 8 (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) = (1𝑜𝑜 𝐵))
2625sseq1d 3611 . . . . . . 7 (𝐴 = 1𝑜 → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (1𝑜𝑜 𝐵) ⊆ 1𝑜))
2724, 26syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2821, 27jaod 395 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2914, 28mtod 189 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30 elpri 4168 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31 df2o3 7518 . . . . 5 2𝑜 = {∅, 1𝑜}
3230, 31eleq2s 2716 . . . 4 (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
3329, 32nsyl 135 . . 3 (𝜑 → ¬ 𝐴 ∈ 2𝑜)
341, 33eldifd 3566 . 2 (𝜑𝐴 ∈ (On ∖ 2𝑜))
3534, 10jca 554 1 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908   ∖ cdif 3552   ⊆ wss 3555  ∅c0 3891  {cpr 4150  {copab 4672  dom cdm 5074  ran crn 5075  Oncon0 5682  ‘cfv 5847  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499   ↑𝑜 coe 7504   CNF ccnf 8502 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-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511 This theorem is referenced by:  cantnflem3  8532  cantnflem4  8533
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