Theorem cardprcALT 36634

Description: Alternate proof of cardprc 10088 from cardprcr 36633. (Contributed by Adrian Ducourtial, 15-Oct-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cardprcALT	⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

Proof of Theorem cardprcALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabab 3484	. . . 4 ⊢ {𝑥 ∈ V ∣ (card‘𝑥) = 𝑥} = {𝑥 ∣ (card‘𝑥) = 𝑥}
2		fveq2 6908	. . . . . 6 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
3		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	2, 3	eqeq12d 2779	. . . . 5 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
5	4	cbvrabv 3426	. . . 4 ⊢ {𝑥 ∈ V ∣ (card‘𝑥) = 𝑥} = {𝑦 ∈ V ∣ (card‘𝑦) = 𝑦}
6	1, 5	eqtr3i 2788	. . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∈ V ∣ (card‘𝑦) = 𝑦}
7		fvexd 6923	. . 3 ⊢ ({𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V → (har‘∪ {𝑥 ∣ (card‘𝑥) = 𝑥}) ∈ V)
8	6, 7	cardprcr 36633	. 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
9	8	nelir 3068	1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741   ∉ wnel 3065  {crab 3416  Vcvv 3454  ∪ cuni 4870  ‘cfv 6558  harchar 9577  cardccrd 10043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-er 8745  df-en 9002  df-dom 9003  df-oi 9531  df-har 9578  df-card 10047

This theorem is used by: (None)