Theorem cardne 9883

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 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)

Proof of Theorem cardne

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6869	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2		cardon 9862	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
3	2	oneli 6433	. . . . . . . . 9 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4		breq1 5089	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
5	4	onintss 6370	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (card‘𝐵) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
7	6	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
8		cardval3 9870	. . . . . . . . 9 ⊢ (𝐵 ∈ dom card → (card‘𝐵) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵})
9	8	sseq1d 3954	. . . . . . . 8 ⊢ (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
11	7, 10	sylibrd 259	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → (card‘𝐵) ⊆ 𝐴))
12		ontri1 6352	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	2, 3, 12	sylancr 588	. . . . . . 7 ⊢ (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
14	13	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
15	11, 14	sylibd 239	. . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
16	15	con2d 134	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
17	16	ex 412	. . 3 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)))
18	17	pm2.43d 53	. 2 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
19	1, 18	mpcom 38	1 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  {crab 3390   ⊆ wss 3890  ∩ cint 4890   class class class wbr 5086  dom cdm 5625  Oncon0 6318  ‘cfv 6493   ≈ cen 8884  cardccrd 9853

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-en 8888  df-card 9857

This theorem is referenced by:  carden2b  9885  cardlim  9890  cardsdomelir  9891