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Theorem elsetrecslem 47230
Description: Lemma for elsetrecs 47231. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 47227. To see why this lemma also requires setrec1 47222, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
elsetrecs.1 𝐵 = setrecs(𝐹)
Assertion
Ref Expression
elsetrecslem (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem elsetrecslem
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ssdifsn 4749 . . . . 5 (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵𝐵 ∧ ¬ 𝐴𝐵))
21simprbi 498 . . . 4 (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴𝐵)
32con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4 elsetrecs.1 . . . 4 𝐵 = setrecs(𝐹)
5 sseq1 3970 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐵𝑎𝐵))
6 fveq2 6843 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
76eleq2d 2820 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴 ∈ (𝐹𝑎)))
85, 7anbi12d 632 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
98notbid 318 . . . . . . 7 (𝑥 = 𝑎 → (¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
109spvv 2001 . . . . . 6 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
11 imnan 401 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
12 idd 24 . . . . . . . . . . 11 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ¬ 𝐴 ∈ (𝐹𝑎)))
13 vex 3448 . . . . . . . . . . . . 13 𝑎 ∈ V
1413a1i 11 . . . . . . . . . . . 12 (𝑎𝐵𝑎 ∈ V)
15 id 22 . . . . . . . . . . . 12 (𝑎𝐵𝑎𝐵)
164, 14, 15setrec1 47222 . . . . . . . . . . 11 (𝑎𝐵 → (𝐹𝑎) ⊆ 𝐵)
1712, 16jctild 527 . . . . . . . . . 10 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1817a2i 14 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1911, 18sylbir 234 . . . . . . . 8 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
2019adantrd 493 . . . . . . 7 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → ((𝑎𝐵 ∧ ¬ 𝐴𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
21 ssdifsn 4749 . . . . . . 7 (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎𝐵 ∧ ¬ 𝐴𝑎))
22 ssdifsn 4749 . . . . . . 7 ((𝐹𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎)))
2320, 21, 223imtr4g 296 . . . . . 6 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2410, 23syl 17 . . . . 5 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2524alrimiv 1931 . . . 4 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
264, 25setrec2v 47227 . . 3 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
273, 26nsyl 140 . 2 (𝐴𝐵 → ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
28 df-ex 1783 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
2927, 28sylibr 233 1 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  cdif 3908  wss 3911  {csn 4587  cfv 6497  setrecscsetrecs 47214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-r1 9705  df-rank 9706  df-setrecs 47215
This theorem is referenced by:  elsetrecs  47231
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