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Theorem infdiffi 9194
Description: Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infdiffi ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)

Proof of Theorem infdiffi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 4007 . . . . . 6 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ∖ ∅))
2 dif0 4261 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
31, 2eqtrdi 2789 . . . . 5 (𝑥 = ∅ → (𝐴𝑥) = 𝐴)
43breq1d 5040 . . . 4 (𝑥 = ∅ → ((𝐴𝑥) ≈ 𝐴𝐴𝐴))
54imbi2d 344 . . 3 (𝑥 = ∅ → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴𝐴𝐴)))
6 difeq2 4007 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
76breq1d 5040 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝑦) ≈ 𝐴))
87imbi2d 344 . . 3 (𝑥 = 𝑦 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴)))
9 difeq2 4007 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ∖ (𝑦 ∪ {𝑧})))
10 difun1 4180 . . . . . 6 (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∖ {𝑧})
119, 10eqtrdi 2789 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = ((𝐴𝑦) ∖ {𝑧}))
1211breq1d 5040 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴𝑥) ≈ 𝐴 ↔ ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
1312imbi2d 344 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
14 difeq2 4007 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
1514breq1d 5040 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ 𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
1615imbi2d 344 . . 3 (𝑥 = 𝐵 → ((ω ≼ 𝐴 → (𝐴𝑥) ≈ 𝐴) ↔ (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴)))
17 reldom 8561 . . . . 5 Rel ≼
1817brrelex2i 5580 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
19 enrefg 8587 . . . 4 (𝐴 ∈ V → 𝐴𝐴)
2018, 19syl 17 . . 3 (ω ≼ 𝐴𝐴𝐴)
21 domen2 8710 . . . . . . . . 9 ((𝐴𝑦) ≈ 𝐴 → (ω ≼ (𝐴𝑦) ↔ ω ≼ 𝐴))
2221biimparc 483 . . . . . . . 8 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ω ≼ (𝐴𝑦))
23 infdifsn 9193 . . . . . . . 8 (ω ≼ (𝐴𝑦) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
2422, 23syl 17 . . . . . . 7 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦))
25 entr 8607 . . . . . . 7 ((((𝐴𝑦) ∖ {𝑧}) ≈ (𝐴𝑦) ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2624, 25sylancom 591 . . . . . 6 ((ω ≼ 𝐴 ∧ (𝐴𝑦) ≈ 𝐴) → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)
2726ex 416 . . . . 5 (ω ≼ 𝐴 → ((𝐴𝑦) ≈ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
2827a2i 14 . . . 4 ((ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴))
2928a1i 11 . . 3 (𝑦 ∈ Fin → ((ω ≼ 𝐴 → (𝐴𝑦) ≈ 𝐴) → (ω ≼ 𝐴 → ((𝐴𝑦) ∖ {𝑧}) ≈ 𝐴)))
305, 8, 13, 16, 20, 29findcard2 8763 . 2 (𝐵 ∈ Fin → (ω ≼ 𝐴 → (𝐴𝐵) ≈ 𝐴))
3130impcom 411 1 ((ω ≼ 𝐴𝐵 ∈ Fin) → (𝐴𝐵) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  cdif 3840  cun 3841  c0 4211  {csn 4516   class class class wbr 5030  ωcom 7599  cen 8552  cdom 8553  Fincfn 8555
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-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-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-er 8320  df-en 8556  df-dom 8557  df-fin 8559
This theorem is referenced by: (None)
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