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Theorem cardne 9956
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Proof of Theorem cardne
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6925 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐡 ∈ dom card)
2 cardon 9935 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
32oneli 6475 . . . . . . . . 9 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 ∈ On)
4 breq1 5150 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐡 ↔ 𝐴 β‰ˆ 𝐡))
54onintss 6412 . . . . . . . . 9 (𝐴 ∈ On β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
76adantl 482 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
8 cardval3 9943 . . . . . . . . 9 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡})
98sseq1d 4012 . . . . . . . 8 (𝐡 ∈ dom card β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
109adantr 481 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
117, 10sylibrd 258 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΅) βŠ† 𝐴))
12 ontri1 6395 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
132, 3, 12sylancr 587 . . . . . . 7 (𝐴 ∈ (cardβ€˜π΅) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1413adantl 482 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1511, 14sylibd 238 . . . . 5 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1615con2d 134 . . . 4 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
1716ex 413 . . 3 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)))
1817pm2.43d 53 . 2 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
191, 18mpcom 38 1 (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  {crab 3432   βŠ† wss 3947  βˆ© cint 4949   class class class wbr 5147  dom cdm 5675  Oncon0 6361  β€˜cfv 6540   β‰ˆ cen 8932  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-en 8936  df-card 9930
This theorem is referenced by:  carden2b  9958  cardlim  9963  cardsdomelir  9964
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