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Theorem cardne 9906
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 6880 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐡 ∈ dom card)
2 cardon 9885 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
32oneli 6432 . . . . . . . . 9 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 ∈ On)
4 breq1 5109 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐡 ↔ 𝐴 β‰ˆ 𝐡))
54onintss 6369 . . . . . . . . 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 483 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
8 cardval3 9893 . . . . . . . . 9 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡})
98sseq1d 3976 . . . . . . . 8 (𝐡 ∈ dom card β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
109adantr 482 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
117, 10sylibrd 259 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΅) βŠ† 𝐴))
12 ontri1 6352 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
132, 3, 12sylancr 588 . . . . . . 7 (𝐴 ∈ (cardβ€˜π΅) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1413adantl 483 . . . . . 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 414 . . 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 397   ∈ wcel 2107  {crab 3406   βŠ† wss 3911  βˆ© cint 4908   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497   β‰ˆ cen 8883  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-en 8887  df-card 9880
This theorem is referenced by:  carden2b  9908  cardlim  9913  cardsdomelir  9914
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