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Theorem unfiOLD 9264
Description: Obsolete version of unfi 9123 as of 7-Aug-2024. (Contributed by NM, 16-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
unfiOLD ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)

Proof of Theorem unfiOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diffi 9130 . 2 (𝐵 ∈ Fin → (𝐵𝐴) ∈ Fin)
2 reeanv 3220 . . . 4 (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴𝑥 ∧ (𝐵𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴𝑥 ∧ ∃𝑦 ∈ ω (𝐵𝐴) ≈ 𝑦))
3 isfi 8923 . . . . 5 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 8923 . . . . 5 ((𝐵𝐴) ∈ Fin ↔ ∃𝑦 ∈ ω (𝐵𝐴) ≈ 𝑦)
53, 4anbi12i 628 . . . 4 ((𝐴 ∈ Fin ∧ (𝐵𝐴) ∈ Fin) ↔ (∃𝑥 ∈ ω 𝐴𝑥 ∧ ∃𝑦 ∈ ω (𝐵𝐴) ≈ 𝑦))
62, 5bitr4i 278 . . 3 (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴𝑥 ∧ (𝐵𝐴) ≈ 𝑦) ↔ (𝐴 ∈ Fin ∧ (𝐵𝐴) ∈ Fin))
7 nnacl 8563 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω)
8 unfilem3 9263 . . . . . . 7 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
9 entr 8953 . . . . . . . 8 (((𝐵𝐴) ≈ 𝑦𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
109expcom 415 . . . . . . 7 (𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥) → ((𝐵𝐴) ≈ 𝑦 → (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)))
118, 10syl 17 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐵𝐴) ≈ 𝑦 → (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)))
12 disjdif 4436 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
13 disjdif 4436 . . . . . . . 8 (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅
14 unen 8997 . . . . . . . 8 (((𝐴𝑥 ∧ (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) ∧ ((𝐴 ∩ (𝐵𝐴)) = ∅ ∧ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅)) → (𝐴 ∪ (𝐵𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
1512, 13, 14mpanr12 704 . . . . . . 7 ((𝐴𝑥 ∧ (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
16 undif2 4441 . . . . . . . . 9 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
1716a1i 11 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵))
18 nnaword1 8581 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +o 𝑦))
19 undif 4446 . . . . . . . . 9 (𝑥 ⊆ (𝑥 +o 𝑦) ↔ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦))
2018, 19sylib 217 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦))
2117, 20breq12d 5123 . . . . . . 7 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ∪ (𝐵𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) ↔ (𝐴𝐵) ≈ (𝑥 +o 𝑦)))
2215, 21imbitrid 243 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑥 ∧ (𝐵𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴𝐵) ≈ (𝑥 +o 𝑦)))
2311, 22sylan2d 606 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑥 ∧ (𝐵𝐴) ≈ 𝑦) → (𝐴𝐵) ≈ (𝑥 +o 𝑦)))
24 breq2 5114 . . . . . . 7 (𝑧 = (𝑥 +o 𝑦) → ((𝐴𝐵) ≈ 𝑧 ↔ (𝐴𝐵) ≈ (𝑥 +o 𝑦)))
2524rspcev 3584 . . . . . 6 (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴𝐵) ≈ (𝑥 +o 𝑦)) → ∃𝑧 ∈ ω (𝐴𝐵) ≈ 𝑧)
26 isfi 8923 . . . . . 6 ((𝐴𝐵) ∈ Fin ↔ ∃𝑧 ∈ ω (𝐴𝐵) ≈ 𝑧)
2725, 26sylibr 233 . . . . 5 (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴𝐵) ≈ (𝑥 +o 𝑦)) → (𝐴𝐵) ∈ Fin)
287, 23, 27syl6an 683 . . . 4 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑥 ∧ (𝐵𝐴) ≈ 𝑦) → (𝐴𝐵) ∈ Fin))
2928rexlimivv 3197 . . 3 (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴𝑥 ∧ (𝐵𝐴) ≈ 𝑦) → (𝐴𝐵) ∈ Fin)
306, 29sylbir 234 . 2 ((𝐴 ∈ Fin ∧ (𝐵𝐴) ∈ Fin) → (𝐴𝐵) ∈ Fin)
311, 30sylan2 594 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3074  cdif 3912  cun 3913  cin 3914  wss 3915  c0 4287   class class class wbr 5110  (class class class)co 7362  ωcom 7807   +o coa 8414  cen 8887  Fincfn 8890
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-en 8891  df-fin 8894
This theorem is referenced by: (None)
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