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Theorem cantnflem2 9131
 Description: Lemma for cantnf 9134. (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 8140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2anc 586 . . . . . . . . 9 (𝜑 → (𝐴o 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴o 𝐵))
6 onelon 6192 . . . . . . . . 9 (((𝐴o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴o 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 586 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 8104 . . . . . . . 8 (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 585 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1o))
1110eldifbd 3926 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1o)
12 ssel 3940 . . . . . . 7 ((𝐴o 𝐵) ⊆ 1o → (𝐶 ∈ (𝐴o 𝐵) → 𝐶 ∈ 1o))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴o 𝐵) ⊆ 1o𝐶 ∈ 1o))
1411, 13mtod 200 . . . . 5 (𝜑 → ¬ (𝐴o 𝐵) ⊆ 1o)
15 oe0m 8121 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑o 𝐵) = (1o𝐵))
17 difss 4087 . . . . . . . 8 (1o𝐵) ⊆ 1o
1816, 17eqsstrdi 4000 . . . . . . 7 (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19 oveq1 7140 . . . . . . . 8 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
2019sseq1d 3977 . . . . . . 7 (𝐴 = ∅ → ((𝐴o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
2118, 20syl5ibrcom 249 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴o 𝐵) ⊆ 1o))
22 oe1m 8149 . . . . . . . 8 (𝐵 ∈ On → (1oo 𝐵) = 1o)
23 eqimss 4002 . . . . . . . 8 ((1oo 𝐵) = 1o → (1oo 𝐵) ⊆ 1o)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1oo 𝐵) ⊆ 1o)
25 oveq1 7140 . . . . . . . 8 (𝐴 = 1o → (𝐴o 𝐵) = (1oo 𝐵))
2625sseq1d 3977 . . . . . . 7 (𝐴 = 1o → ((𝐴o 𝐵) ⊆ 1o ↔ (1oo 𝐵) ⊆ 1o))
2724, 26syl5ibrcom 249 . . . . . 6 (𝜑 → (𝐴 = 1o → (𝐴o 𝐵) ⊆ 1o))
2821, 27jaod 855 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴o 𝐵) ⊆ 1o))
2914, 28mtod 200 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o))
30 elpri 4565 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31 df2o3 8095 . . . . 5 2o = {∅, 1o}
3230, 31eleq2s 2929 . . . 4 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
3329, 32nsyl 142 . . 3 (𝜑 → ¬ 𝐴 ∈ 2o)
341, 33eldifd 3924 . 2 (𝜑𝐴 ∈ (On ∖ 2o))
3534, 10jca 514 1 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∀wral 3125  ∃wrex 3126   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4269  {cpr 4545  {copab 5104  dom cdm 5531  ran crn 5532  Oncon0 6167  ‘cfv 6331  (class class class)co 7133  1oc1o 8073  2oc2o 8074   ↑o coe 8079   CNF ccnf 9102 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-rep 5166  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-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  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-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  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-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-omul 8085  df-oexp 8086 This theorem is referenced by:  cantnflem3  9132  cantnflem4  9133
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