MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflem2 Structured version   Visualization version   GIF version

Theorem cantnflem2 9318
Description: Lemma for cantnf 9321. (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 8273 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐴o 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴o 𝐵))
6 onelon 6247 . . . . . . . . 9 (((𝐴o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴o 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 587 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 8237 . . . . . . . 8 (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 586 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1o))
1110eldifbd 3888 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1o)
12 ssel 3902 . . . . . . 7 ((𝐴o 𝐵) ⊆ 1o → (𝐶 ∈ (𝐴o 𝐵) → 𝐶 ∈ 1o))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴o 𝐵) ⊆ 1o𝐶 ∈ 1o))
1411, 13mtod 201 . . . . 5 (𝜑 → ¬ (𝐴o 𝐵) ⊆ 1o)
15 oe0m 8254 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑o 𝐵) = (1o𝐵))
17 difss 4055 . . . . . . . 8 (1o𝐵) ⊆ 1o
1816, 17eqsstrdi 3964 . . . . . . 7 (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19 oveq1 7229 . . . . . . . 8 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
2019sseq1d 3941 . . . . . . 7 (𝐴 = ∅ → ((𝐴o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
2118, 20syl5ibrcom 250 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴o 𝐵) ⊆ 1o))
22 oe1m 8282 . . . . . . . 8 (𝐵 ∈ On → (1oo 𝐵) = 1o)
23 eqimss 3966 . . . . . . . 8 ((1oo 𝐵) = 1o → (1oo 𝐵) ⊆ 1o)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1oo 𝐵) ⊆ 1o)
25 oveq1 7229 . . . . . . . 8 (𝐴 = 1o → (𝐴o 𝐵) = (1oo 𝐵))
2625sseq1d 3941 . . . . . . 7 (𝐴 = 1o → ((𝐴o 𝐵) ⊆ 1o ↔ (1oo 𝐵) ⊆ 1o))
2724, 26syl5ibrcom 250 . . . . . 6 (𝜑 → (𝐴 = 1o → (𝐴o 𝐵) ⊆ 1o))
2821, 27jaod 859 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴o 𝐵) ⊆ 1o))
2914, 28mtod 201 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o))
30 elpri 4572 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31 df2o3 8226 . . . . 5 2o = {∅, 1o}
3230, 31eleq2s 2857 . . . 4 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
3329, 32nsyl 142 . . 3 (𝜑 → ¬ 𝐴 ∈ 2o)
341, 33eldifd 3886 . 2 (𝜑𝐴 ∈ (On ∖ 2o))
3534, 10jca 515 1 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 847   = wceq 1543  wcel 2111  wral 3062  wrex 3063  cdif 3872  wss 3875  c0 4246  {cpr 4552  {copab 5124  dom cdm 5560  ran crn 5561  Oncon0 6222  cfv 6389  (class class class)co 7222  1oc1o 8204  2oc2o 8205  o coe 8210   CNF ccnf 9289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-pss 3894  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-tr 5171  df-id 5464  df-eprel 5469  df-po 5477  df-so 5478  df-fr 5518  df-we 5520  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-pred 6169  df-ord 6225  df-on 6226  df-lim 6227  df-suc 6228  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-om 7654  df-wrecs 8056  df-recs 8117  df-rdg 8155  df-1o 8211  df-2o 8212  df-oadd 8215  df-omul 8216  df-oexp 8217
This theorem is referenced by:  cantnflem3  9319  cantnflem4  9320
  Copyright terms: Public domain W3C validator