Theorem infunsdom1 8918

Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 8919 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
infunsdom1	⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Proof of Theorem infunsdom1

Step	Hyp	Ref	Expression
1		simprl 790	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐴 ≼ 𝐵)
2		domsdomtr 7980	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
3	1, 2	sylan 487	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → 𝐴 ≺ ω)
4		unfi2 8114	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
5	3, 4	sylancom 698	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
6		simpllr 795	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → ω ≼ 𝑋)
7		sdomdomtr 7978	. . 3 ⊢ (((𝐴 ∪ 𝐵) ≺ ω ∧ ω ≼ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
8	5, 6, 7	syl2anc 691	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
9		omelon 8426	. . . . . 6 ⊢ ω ∈ On
10		onenon 8658	. . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ω ∈ dom card
12		simpll 786	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝑋 ∈ dom card)
13		sdomdom 7869	. . . . . . 7 ⊢ (𝐵 ≺ 𝑋 → 𝐵 ≼ 𝑋)
14	13	ad2antll 761	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ≼ 𝑋)
15		numdom 8744	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ 𝐵 ≼ 𝑋) → 𝐵 ∈ dom card)
16	12, 14, 15	syl2anc 691	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → 𝐵 ∈ dom card)
17		domtri2 8698	. . . . 5 ⊢ ((ω ∈ dom card ∧ 𝐵 ∈ dom card) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
18	11, 16, 17	sylancr 694	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (ω ≼ 𝐵 ↔ ¬ 𝐵 ≺ ω))
19	18	biimpar 501	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → ω ≼ 𝐵)
20		uncom 3719	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
21	16	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ∈ dom card)
22		simpr 476	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → ω ≼ 𝐵)
23	1	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐴 ≼ 𝐵)
24		infunabs 8912	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
25	21, 22, 23, 24	syl3anc 1318	. . . . 5 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
26	20, 25	syl5eqbr 4618	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
27		simplrr 797	. . . 4 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → 𝐵 ≺ 𝑋)
28		ensdomtr 7981	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐵 ∧ 𝐵 ≺ 𝑋) → (𝐴 ∪ 𝐵) ≺ 𝑋)
29	26, 27, 28	syl2anc 691	. . 3 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ω ≼ 𝐵) → (𝐴 ∪ 𝐵) ≺ 𝑋)
30	19, 29	syldan 486	. 2 ⊢ ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) ∧ ¬ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ 𝑋)
31	8, 30	pm2.61dan 828	1 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ∪ cun 3538   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873

This theorem is referenced by:  infunsdom  8919