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Theorem elsetrecslem 47831
Description: Lemma for elsetrecs 47832. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 47828. To see why this lemma also requires setrec1 47823, 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 4790 . . . . 5 (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵𝐵 ∧ ¬ 𝐴𝐵))
21simprbi 495 . . . 4 (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴𝐵)
32con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4 elsetrecs.1 . . . 4 𝐵 = setrecs(𝐹)
5 sseq1 4006 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐵𝑎𝐵))
6 fveq2 6890 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
76eleq2d 2817 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴 ∈ (𝐹𝑎)))
85, 7anbi12d 629 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
98notbid 317 . . . . . . 7 (𝑥 = 𝑎 → (¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
109spvv 1998 . . . . . 6 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
11 imnan 398 . . . . . . . . 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 47823 . . . . . . . . . . 11 (𝑎𝐵 → (𝐹𝑎) ⊆ 𝐵)
1712, 16jctild 524 . . . . . . . . . 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 490 . . . . . . 7 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → ((𝑎𝐵 ∧ ¬ 𝐴𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
21 ssdifsn 4790 . . . . . . 7 (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎𝐵 ∧ ¬ 𝐴𝑎))
22 ssdifsn 4790 . . . . . . 7 ((𝐹𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎)))
2320, 21, 223imtr4g 295 . . . . . 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 47828 . . 3 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
273, 26nsyl 140 . 2 (𝐴𝐵 → ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
28 df-ex 1780 . 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 394  wal 1537   = wceq 1539  wex 1779  wcel 2104  Vcvv 3472  cdif 3944  wss 3947  {csn 4627  cfv 6542  setrecscsetrecs 47815
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762  df-setrecs 47816
This theorem is referenced by:  elsetrecs  47832
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