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Theorem unfilem1 8109
 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 6967 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 703 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 7586 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
61, 4, 5mp3an23 1408 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
87ibi 255 . . . . . . 7 (𝑥𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9 nnaword1 7596 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10 nnord 6965 . . . . . . . . . . 11 (𝐴 ∈ ω → Ord 𝐴)
114, 10ax-mp 5 . . . . . . . . . 10 Ord 𝐴
12 nnacl 7578 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13 nnord 6965 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
1412, 13syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +𝑜 𝑥))
15 ordtri1 5673 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
1611, 14, 15sylancr 694 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
179, 16mpbid 221 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
184, 3, 17sylancr 694 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
198, 18jca 553 . . . . . 6 (𝑥𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20 eleq1 2676 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21 eleq1 2676 . . . . . . . . 9 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2221notbid 307 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2320, 22anbi12d 743 . . . . . . 7 (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
2423biimparc 503 . . . . . 6 ((((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2519, 24sylan 487 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2625rexlimiva 3010 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
274, 1nnacli 7581 . . . . . . . 8 (𝐴 +𝑜 𝐵) ∈ ω
28 elnn 6967 . . . . . . . 8 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
2927, 28mpan2 703 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30 nnord 6965 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
31 ordtri1 5673 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3210, 30, 31syl2an 493 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
33 nnawordex 7604 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
3432, 33bitr3d 269 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
354, 29, 34sylancr 694 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36 eleq1 2676 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
376, 36sylan9bb 732 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 ∈ (𝐴 +𝑜 𝐵)))
3837biimprcd 239 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥𝐵))
39 eqcom 2617 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4039biimpi 205 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4140adantl 481 . . . . . . . . 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 554 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥))))
4443reximdv2 2997 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4535, 44sylbid 229 . . . . 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 198 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
48 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
49 ovex 6577 . . . 4 (𝐴 +𝑜 𝑥) ∈ V
5048, 49elrnmpti 5297 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51 eldif 3550 . . 3 (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
5247, 50, 513bitr4i 291 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
5352eqriv 2607 1 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540   ↦ cmpt 4643  ran crn 5039  Ord word 5639  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451 This theorem is referenced by:  unfilem2  8110
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