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Theorem unfilem1 8377
Description: Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
unfilem1.1 𝐴 ∈ ω
unfilem1.2 𝐵 ∈ ω
unfilem1.3 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
Assertion
Ref Expression
unfilem1 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unfilem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 unfilem1.2 . . . . . . . . . 10 𝐵 ∈ ω
2 elnn 7228 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 709 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 7856 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
61, 4, 5mp3an23 1553 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
87ibi 256 . . . . . . 7 (𝑥𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9 nnaword1 7866 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10 nnord 7226 . . . . . . . . . . 11 (𝐴 ∈ ω → Ord 𝐴)
114, 10ax-mp 5 . . . . . . . . . 10 Ord 𝐴
12 nnacl 7848 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13 nnord 7226 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
1412, 13syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +𝑜 𝑥))
15 ordtri1 5905 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
1611, 14, 15sylancr 698 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
179, 16mpbid 222 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
184, 3, 17sylancr 698 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
198, 18jca 555 . . . . . 6 (𝑥𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20 eleq1 2815 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21 eleq1 2815 . . . . . . . . 9 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2221notbid 307 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2320, 22anbi12d 749 . . . . . . 7 (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
2423biimparc 505 . . . . . 6 ((((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2519, 24sylan 489 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2625rexlimiva 3154 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
274, 1nnacli 7851 . . . . . . . 8 (𝐴 +𝑜 𝐵) ∈ ω
28 elnn 7228 . . . . . . . 8 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
2927, 28mpan2 709 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30 nnord 7226 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
31 ordtri1 5905 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3210, 30, 31syl2an 495 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
33 nnawordex 7874 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
3432, 33bitr3d 270 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
354, 29, 34sylancr 698 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36 eleq1 2815 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
376, 36sylan9bb 738 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 ∈ (𝐴 +𝑜 𝐵)))
3837biimprcd 240 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥𝐵))
39 eqcom 2755 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4039biimpi 206 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4140adantl 473 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥))
4241a1i 11 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥)))
4338, 42jcad 556 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥))))
4443reximdv2 3140 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4535, 44sylbid 230 . . . . 5 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4645imp 444 . . . 4 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
4726, 46impbii 199 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
48 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
49 ovex 6829 . . . 4 (𝐴 +𝑜 𝑥) ∈ V
5048, 49elrnmpti 5519 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51 eldif 3713 . . 3 (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
5247, 50, 513bitr4i 292 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
5352eqriv 2745 1 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wrex 3039  cdif 3700  wss 3703  cmpt 4869  ran crn 5255  Ord word 5871  (class class class)co 6801  ωcom 7218   +𝑜 coa 7714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-oadd 7721
This theorem is referenced by:  unfilem2  8378
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