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Theorem djuinf 9688
Description: A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
djuinf (ω ≼ 𝐴 ↔ ω ≼ (𝐴𝐴))

Proof of Theorem djuinf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 reldom 8561 . . . . 5 Rel ≼
21brrelex2i 5580 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 djudoml 9684 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴𝐴))
42, 2, 3syl2anc 587 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴𝐴))
5 domtr 8608 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴𝐴)) → ω ≼ (𝐴𝐴))
64, 5mpdan 687 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴𝐴))
71brrelex2i 5580 . . . 4 (ω ≼ (𝐴𝐴) → (𝐴𝐴) ∈ V)
8 anidm 568 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 ∈ V) ↔ 𝐴 ∈ V)
9 djuexb 9411 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 ∈ V) ↔ (𝐴𝐴) ∈ V)
108, 9bitr3i 280 . . . 4 (𝐴 ∈ V ↔ (𝐴𝐴) ∈ V)
117, 10sylibr 237 . . 3 (ω ≼ (𝐴𝐴) → 𝐴 ∈ V)
12 domeng 8569 . . . . 5 ((𝐴𝐴) ∈ V → (ω ≼ (𝐴𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴))))
137, 12syl 17 . . . 4 (ω ≼ (𝐴𝐴) → (ω ≼ (𝐴𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴))))
1413ibi 270 . . 3 (ω ≼ (𝐴𝐴) → ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)))
15 indi 4164 . . . . . . 7 (𝑥 ∩ (({∅} × 𝐴) ∪ ({1o} × 𝐴))) = ((𝑥 ∩ ({∅} × 𝐴)) ∪ (𝑥 ∩ ({1o} × 𝐴)))
16 simpr 488 . . . . . . . . 9 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → 𝑥 ⊆ (𝐴𝐴))
17 df-dju 9403 . . . . . . . . 9 (𝐴𝐴) = (({∅} × 𝐴) ∪ ({1o} × 𝐴))
1816, 17sseqtrdi 3927 . . . . . . . 8 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → 𝑥 ⊆ (({∅} × 𝐴) ∪ ({1o} × 𝐴)))
19 df-ss 3860 . . . . . . . 8 (𝑥 ⊆ (({∅} × 𝐴) ∪ ({1o} × 𝐴)) ↔ (𝑥 ∩ (({∅} × 𝐴) ∪ ({1o} × 𝐴))) = 𝑥)
2018, 19sylib 221 . . . . . . 7 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → (𝑥 ∩ (({∅} × 𝐴) ∪ ({1o} × 𝐴))) = 𝑥)
2115, 20eqtr3id 2787 . . . . . 6 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → ((𝑥 ∩ ({∅} × 𝐴)) ∪ (𝑥 ∩ ({1o} × 𝐴))) = 𝑥)
22 ensym 8604 . . . . . . 7 (ω ≈ 𝑥𝑥 ≈ ω)
2322adantr 484 . . . . . 6 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → 𝑥 ≈ ω)
2421, 23eqbrtrd 5052 . . . . 5 ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → ((𝑥 ∩ ({∅} × 𝐴)) ∪ (𝑥 ∩ ({1o} × 𝐴))) ≈ ω)
25 cdainflem 9687 . . . . . 6 (((𝑥 ∩ ({∅} × 𝐴)) ∪ (𝑥 ∩ ({1o} × 𝐴))) ≈ ω → ((𝑥 ∩ ({∅} × 𝐴)) ≈ ω ∨ (𝑥 ∩ ({1o} × 𝐴)) ≈ ω))
26 snex 5298 . . . . . . . . . . 11 {∅} ∈ V
27 xpexg 7491 . . . . . . . . . . 11 (({∅} ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ∈ V)
2826, 27mpan 690 . . . . . . . . . 10 (𝐴 ∈ V → ({∅} × 𝐴) ∈ V)
29 inss2 4120 . . . . . . . . . 10 (𝑥 ∩ ({∅} × 𝐴)) ⊆ ({∅} × 𝐴)
30 ssdomg 8601 . . . . . . . . . 10 (({∅} × 𝐴) ∈ V → ((𝑥 ∩ ({∅} × 𝐴)) ⊆ ({∅} × 𝐴) → (𝑥 ∩ ({∅} × 𝐴)) ≼ ({∅} × 𝐴)))
3128, 29, 30mpisyl 21 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ ({∅} × 𝐴)) ≼ ({∅} × 𝐴))
32 0ex 5175 . . . . . . . . . 10 ∅ ∈ V
33 xpsnen2g 8659 . . . . . . . . . 10 ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴)
3432, 33mpan 690 . . . . . . . . 9 (𝐴 ∈ V → ({∅} × 𝐴) ≈ 𝐴)
35 domentr 8614 . . . . . . . . 9 (((𝑥 ∩ ({∅} × 𝐴)) ≼ ({∅} × 𝐴) ∧ ({∅} × 𝐴) ≈ 𝐴) → (𝑥 ∩ ({∅} × 𝐴)) ≼ 𝐴)
3631, 34, 35syl2anc 587 . . . . . . . 8 (𝐴 ∈ V → (𝑥 ∩ ({∅} × 𝐴)) ≼ 𝐴)
37 domen1 8709 . . . . . . . 8 ((𝑥 ∩ ({∅} × 𝐴)) ≈ ω → ((𝑥 ∩ ({∅} × 𝐴)) ≼ 𝐴 ↔ ω ≼ 𝐴))
3836, 37syl5ibcom 248 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ∩ ({∅} × 𝐴)) ≈ ω → ω ≼ 𝐴))
39 snex 5298 . . . . . . . . . . 11 {1o} ∈ V
40 xpexg 7491 . . . . . . . . . . 11 (({1o} ∈ V ∧ 𝐴 ∈ V) → ({1o} × 𝐴) ∈ V)
4139, 40mpan 690 . . . . . . . . . 10 (𝐴 ∈ V → ({1o} × 𝐴) ∈ V)
42 inss2 4120 . . . . . . . . . 10 (𝑥 ∩ ({1o} × 𝐴)) ⊆ ({1o} × 𝐴)
43 ssdomg 8601 . . . . . . . . . 10 (({1o} × 𝐴) ∈ V → ((𝑥 ∩ ({1o} × 𝐴)) ⊆ ({1o} × 𝐴) → (𝑥 ∩ ({1o} × 𝐴)) ≼ ({1o} × 𝐴)))
4441, 42, 43mpisyl 21 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ ({1o} × 𝐴)) ≼ ({1o} × 𝐴))
45 1on 8138 . . . . . . . . . 10 1o ∈ On
46 xpsnen2g 8659 . . . . . . . . . 10 ((1o ∈ On ∧ 𝐴 ∈ V) → ({1o} × 𝐴) ≈ 𝐴)
4745, 46mpan 690 . . . . . . . . 9 (𝐴 ∈ V → ({1o} × 𝐴) ≈ 𝐴)
48 domentr 8614 . . . . . . . . 9 (((𝑥 ∩ ({1o} × 𝐴)) ≼ ({1o} × 𝐴) ∧ ({1o} × 𝐴) ≈ 𝐴) → (𝑥 ∩ ({1o} × 𝐴)) ≼ 𝐴)
4944, 47, 48syl2anc 587 . . . . . . . 8 (𝐴 ∈ V → (𝑥 ∩ ({1o} × 𝐴)) ≼ 𝐴)
50 domen1 8709 . . . . . . . 8 ((𝑥 ∩ ({1o} × 𝐴)) ≈ ω → ((𝑥 ∩ ({1o} × 𝐴)) ≼ 𝐴 ↔ ω ≼ 𝐴))
5149, 50syl5ibcom 248 . . . . . . 7 (𝐴 ∈ V → ((𝑥 ∩ ({1o} × 𝐴)) ≈ ω → ω ≼ 𝐴))
5238, 51jaod 858 . . . . . 6 (𝐴 ∈ V → (((𝑥 ∩ ({∅} × 𝐴)) ≈ ω ∨ (𝑥 ∩ ({1o} × 𝐴)) ≈ ω) → ω ≼ 𝐴))
5325, 52syl5 34 . . . . 5 (𝐴 ∈ V → (((𝑥 ∩ ({∅} × 𝐴)) ∪ (𝑥 ∩ ({1o} × 𝐴))) ≈ ω → ω ≼ 𝐴))
5424, 53syl5 34 . . . 4 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → ω ≼ 𝐴))
5554exlimdv 1940 . . 3 (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴𝐴)) → ω ≼ 𝐴))
5611, 14, 55sylc 65 . 2 (ω ≼ (𝐴𝐴) → ω ≼ 𝐴)
576, 56impbii 212 1 (ω ≼ 𝐴 ↔ ω ≼ (𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 846   = wceq 1542  wex 1786  wcel 2114  Vcvv 3398  cun 3841  cin 3842  wss 3843  c0 4211  {csn 4516   class class class wbr 5030   × cxp 5523  Oncon0 6172  ωcom 7599  1oc1o 8124  cen 8552  cdom 8553  cdju 9400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-dju 9403
This theorem is referenced by:  infdif  9709
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