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Theorem inf3lemd 9664
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 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 9662 . . . . 5 (𝐹‘∅) = ∅
71, 6eqtrdi 2790 . . . 4 (𝐴 = ∅ → (𝐹𝐴) = ∅)
8 0ss 4405 . . . 4 ∅ ⊆ 𝑥
97, 8eqsstrdi 4049 . . 3 (𝐴 = ∅ → (𝐹𝐴) ⊆ 𝑥)
109a1d 25 . 2 (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
11 nnsuc 7904 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣)
12 vex 3481 . . . . . . . . . 10 𝑣 ∈ V
132, 3, 12, 5inf3lemc 9663 . . . . . . . . 9 (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹𝑣)))
1413eleq2d 2824 . . . . . . . 8 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹𝑣))))
15 vex 3481 . . . . . . . . . 10 𝑢 ∈ V
16 fvex 6919 . . . . . . . . . 10 (𝐹𝑣) ∈ V
172, 3, 15, 16inf3lema 9661 . . . . . . . . 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 4000 . . . . . 6 (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥)
21 fveq2 6906 . . . . . . 7 (𝐴 = suc 𝑣 → (𝐹𝐴) = (𝐹‘suc 𝑣))
2221sseq1d 4026 . . . . . 6 (𝐴 = suc 𝑣 → ((𝐹𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥))
2320, 22syl5ibrcom 247 . . . . 5 (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹𝐴) ⊆ 𝑥))
2423rexlimiv 3145 . . . 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 3022 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  {crab 3432  Vcvv 3477  cin 3961  wss 3962  c0 4338  cmpt 5230  cres 5690  suc csuc 6387  cfv 6562  ωcom 7886  reccrdg 8447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448
This theorem is referenced by:  inf3lem2  9666  inf3lem3  9667  inf3lem6  9670
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