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Theorem inf3lemd 9621
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9629 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐵 ∈ V
Assertion
Ref Expression
inf3lemd (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)
Distinct variable group:   𝑥,𝑦,𝑤
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem inf3lemd
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . 5 (𝐴 = ∅ → (𝐹𝐴) = (𝐹‘∅))
2 inf3lem.1 . . . . . 6 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
3 inf3lem.2 . . . . . 6 𝐹 = (rec(𝐺, ∅) ↾ ω)
4 inf3lem.3 . . . . . 6 𝐴 ∈ V
5 inf3lem.4 . . . . . 6 𝐵 ∈ V
62, 3, 4, 5inf3lemb 9619 . . . . 5 (𝐹‘∅) = ∅
71, 6eqtrdi 2788 . . . 4 (𝐴 = ∅ → (𝐹𝐴) = ∅)
8 0ss 4396 . . . 4 ∅ ⊆ 𝑥
97, 8eqsstrdi 4036 . . 3 (𝐴 = ∅ → (𝐹𝐴) ⊆ 𝑥)
109a1d 25 . 2 (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
11 nnsuc 7872 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣)
12 vex 3478 . . . . . . . . . 10 𝑣 ∈ V
132, 3, 12, 5inf3lemc 9620 . . . . . . . . 9 (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹𝑣)))
1413eleq2d 2819 . . . . . . . 8 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹𝑣))))
15 vex 3478 . . . . . . . . . 10 𝑢 ∈ V
16 fvex 6904 . . . . . . . . . 10 (𝐹𝑣) ∈ V
172, 3, 15, 16inf3lema 9618 . . . . . . . . 9 (𝑢 ∈ (𝐺‘(𝐹𝑣)) ↔ (𝑢𝑥 ∧ (𝑢𝑥) ⊆ (𝐹𝑣)))
1817simplbi 498 . . . . . . . 8 (𝑢 ∈ (𝐺‘(𝐹𝑣)) → 𝑢𝑥)
1914, 18syl6bi 252 . . . . . . 7 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢𝑥))
2019ssrdv 3988 . . . . . 6 (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥)
21 fveq2 6891 . . . . . . 7 (𝐴 = suc 𝑣 → (𝐹𝐴) = (𝐹‘suc 𝑣))
2221sseq1d 4013 . . . . . 6 (𝐴 = suc 𝑣 → ((𝐹𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥))
2320, 22syl5ibrcom 246 . . . . 5 (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹𝐴) ⊆ 𝑥))
2423rexlimiv 3148 . . . 4 (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹𝐴) ⊆ 𝑥)
2511, 24syl 17 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹𝐴) ⊆ 𝑥)
2625expcom 414 . 2 (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
2710, 26pm2.61ine 3025 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  Vcvv 3474  cin 3947  wss 3948  c0 4322  cmpt 5231  cres 5678  suc csuc 6366  cfv 6543  ωcom 7854  reccrdg 8408
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 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409
This theorem is referenced by:  inf3lem2  9623  inf3lem3  9624  inf3lem6  9627
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