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Theorem omsindsOLD 7873
Description: Obsolete version of omsinds 7872 as of 16-Oct-2024. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
omsinds.1 (𝑥 = 𝑦 → (𝜑𝜓))
omsinds.2 (𝑥 = 𝐴 → (𝜑𝜒))
omsinds.3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
Assertion
Ref Expression
omsindsOLD (𝐴 ∈ ω → 𝜒)
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem omsindsOLD
StepHypRef Expression
1 omsson 7855 . . 3 ω ⊆ On
2 epweon 7758 . . 3 E We On
3 wess 5662 . . 3 (ω ⊆ On → ( E We On → E We ω))
41, 2, 3mp2 9 . 2 E We ω
5 epse 5658 . 2 E Se ω
6 omsinds.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
7 omsinds.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
8 predep 6328 . . . . 5 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥))
9 ordom 7861 . . . . . . 7 Ord ω
10 ordtr 6375 . . . . . . 7 (Ord ω → Tr ω)
11 trss 5275 . . . . . . 7 (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω))
129, 10, 11mp2b 10 . . . . . 6 (𝑥 ∈ ω → 𝑥 ⊆ ω)
13 sseqin2 4214 . . . . . 6 (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥)
1412, 13sylib 217 . . . . 5 (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥)
158, 14eqtrd 2772 . . . 4 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥)
1615raleqdv 3325 . . 3 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦𝑥 𝜓))
17 omsinds.3 . . 3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
1816, 17sylbid 239 . 2 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓𝜑))
194, 5, 6, 7, 18wfis3 6359 1 (𝐴 ∈ ω → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3061  cin 3946  wss 3947  Tr wtr 5264   E cep 5578   We wwe 5629  Predcpred 6296  Ord word 6360  Oncon0 6361  ωcom 7851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-om 7852
This theorem is referenced by: (None)
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