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Theorem elsetrecslem 44988
Description: Lemma for elsetrecs 44989. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 44986. To see why this lemma also requires setrec1 44981, 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 4696 . . . . 5 (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵𝐵 ∧ ¬ 𝐴𝐵))
21simprbi 499 . . . 4 (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴𝐵)
32con2i 141 . . 3 (𝐴𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4 elsetrecs.1 . . . 4 𝐵 = setrecs(𝐹)
5 sseq1 3971 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐵𝑎𝐵))
6 fveq2 6646 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
76eleq2d 2896 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴 ∈ (𝐹𝑎)))
85, 7anbi12d 632 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
98notbid 320 . . . . . . 7 (𝑥 = 𝑎 → (¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
109spvv 2003 . . . . . 6 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
11 imnan 402 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
12 idd 24 . . . . . . . . . . 11 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ¬ 𝐴 ∈ (𝐹𝑎)))
13 vex 3476 . . . . . . . . . . . . 13 𝑎 ∈ V
1413a1i 11 . . . . . . . . . . . 12 (𝑎𝐵𝑎 ∈ V)
15 id 22 . . . . . . . . . . . 12 (𝑎𝐵𝑎𝐵)
164, 14, 15setrec1 44981 . . . . . . . . . . 11 (𝑎𝐵 → (𝐹𝑎) ⊆ 𝐵)
1712, 16jctild 528 . . . . . . . . . 10 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1817a2i 14 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1911, 18sylbir 237 . . . . . . . 8 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
2019adantrd 494 . . . . . . 7 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → ((𝑎𝐵 ∧ ¬ 𝐴𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
21 ssdifsn 4696 . . . . . . 7 (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎𝐵 ∧ ¬ 𝐴𝑎))
22 ssdifsn 4696 . . . . . . 7 ((𝐹𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎)))
2320, 21, 223imtr4g 298 . . . . . 6 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2410, 23syl 17 . . . . 5 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2524alrimiv 1928 . . . 4 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
264, 25setrec2v 44986 . . 3 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
273, 26nsyl 142 . 2 (𝐴𝐵 → ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
28 df-ex 1781 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
2927, 28sylibr 236 1 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1535   = wceq 1537  wex 1780  wcel 2114  Vcvv 3473  cdif 3910  wss 3913  {csn 4543  cfv 6331  setrecscsetrecs 44973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-reg 9034  ax-inf2 9082
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-om 7559  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-r1 9171  df-rank 9172  df-setrecs 44974
This theorem is referenced by:  elsetrecs  44989
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