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Theorem inf3lemd 9667
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9675 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 6906 . . . . 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 9665 . . . . 5 (𝐹‘∅) = ∅
71, 6eqtrdi 2793 . . . 4 (𝐴 = ∅ → (𝐹𝐴) = ∅)
8 0ss 4400 . . . 4 ∅ ⊆ 𝑥
97, 8eqsstrdi 4028 . . 3 (𝐴 = ∅ → (𝐹𝐴) ⊆ 𝑥)
109a1d 25 . 2 (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
11 nnsuc 7905 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣)
12 vex 3484 . . . . . . . . . 10 𝑣 ∈ V
132, 3, 12, 5inf3lemc 9666 . . . . . . . . 9 (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹𝑣)))
1413eleq2d 2827 . . . . . . . 8 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹𝑣))))
15 vex 3484 . . . . . . . . . 10 𝑢 ∈ V
16 fvex 6919 . . . . . . . . . 10 (𝐹𝑣) ∈ V
172, 3, 15, 16inf3lema 9664 . . . . . . . . 9 (𝑢 ∈ (𝐺‘(𝐹𝑣)) ↔ (𝑢𝑥 ∧ (𝑢𝑥) ⊆ (𝐹𝑣)))
1817simplbi 497 . . . . . . . 8 (𝑢 ∈ (𝐺‘(𝐹𝑣)) → 𝑢𝑥)
1914, 18biimtrdi 253 . . . . . . 7 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢𝑥))
2019ssrdv 3989 . . . . . 6 (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥)
21 fveq2 6906 . . . . . . 7 (𝐴 = suc 𝑣 → (𝐹𝐴) = (𝐹‘suc 𝑣))
2221sseq1d 4015 . . . . . 6 (𝐴 = suc 𝑣 → ((𝐹𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥))
2320, 22syl5ibrcom 247 . . . . 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 413 . 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 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  c0 4333  cmpt 5225  cres 5687  suc csuc 6386  cfv 6561  ωcom 7887  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  inf3lem2  9669  inf3lem3  9670  inf3lem6  9673
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