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Theorem unfilem1 8473
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 7315 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 674 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 7946 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
61, 4, 5mp3an23 1570 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
87ibi 258 . . . . . . 7 (𝑥𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9 nnaword1 7956 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10 nnord 7313 . . . . . . . . . . 11 (𝐴 ∈ ω → Ord 𝐴)
114, 10ax-mp 5 . . . . . . . . . 10 Ord 𝐴
12 nnacl 7938 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13 nnord 7313 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
1412, 13syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +𝑜 𝑥))
15 ordtri1 5983 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
1611, 14, 15sylancr 577 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
179, 16mpbid 223 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
184, 3, 17sylancr 577 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
198, 18jca 503 . . . . . 6 (𝑥𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20 eleq1 2884 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21 eleq1 2884 . . . . . . . . 9 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2221notbid 309 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2320, 22anbi12d 618 . . . . . . 7 (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
2423biimparc 467 . . . . . 6 ((((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2519, 24sylan 571 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2625rexlimiva 3227 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
274, 1nnacli 7941 . . . . . . . 8 (𝐴 +𝑜 𝐵) ∈ ω
28 elnn 7315 . . . . . . . 8 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
2927, 28mpan2 674 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30 nnord 7313 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
31 ordtri1 5983 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3210, 30, 31syl2an 585 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
33 nnawordex 7964 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
3432, 33bitr3d 272 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
354, 29, 34sylancr 577 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36 eleq1 2884 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
376, 36sylan9bb 501 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 ∈ (𝐴 +𝑜 𝐵)))
3837biimprcd 241 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥𝐵))
39 eqcom 2824 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4039biimpi 207 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4140adantl 469 . . . . . . . . 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 504 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥))))
4443reximdv2 3212 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4535, 44sylbid 231 . . . . 5 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4645imp 395 . . . 4 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
4726, 46impbii 200 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
48 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
49 ovex 6916 . . . 4 (𝐴 +𝑜 𝑥) ∈ V
5048, 49elrnmpti 5591 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51 eldif 3790 . . 3 (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
5247, 50, 513bitr4i 294 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
5352eqriv 2814 1 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wrex 3108  cdif 3777  wss 3780  cmpt 4934  ran crn 5325  Ord word 5949  (class class class)co 6884  ωcom 7305   +𝑜 coa 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-oadd 7810
This theorem is referenced by:  unfilem2  8474
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